TYPE I MULTIPLIER REPRESENTATIONS

OF LOCALLY COMPACT GROUPS

A.K. H0LZHERR (B.Sc.)

A Thesis submitted for the Degree of

Doctor of PhilosophY

in the University of Adelaide

Department of Pure Mathematici

FebruarY, l99z

by

CONTENTS

SUMMARY

SIGNED STATEMENT

ACKNOt,\lLEDGEMENTS

CHAPTIR I - PRELIMINARIES

Locally compact abel'ian grouPs

Concrete C*-algebras and von Neumann algebras

Classification of von Neumann algebras

Type I, von Neumann algebras and polynomial identities

5. Representations

6. Multip'liers and multipf ier representations

7. Moore groups and the regular representation

B. Induced representations, Mackey's construction

CIIAPTER II - ON THE STRUCTURE OF I¡-TYPE I LOCALLY COMPAüT

ABTLIAN GROUPS

1. Notation and e'lementarY facts

2. Some groups don't admit multipìiers

3. Sonle useful resul ts

4. The connected component of G as a direct sumrnand

5. Local direct products and div'isjble groups

CHAPTER III - MULTIPLIER REPRESENIATIONS OF DISCRETE GROUPS

1. A representations of elements in V(G,t¡)

2. Discrete finite class groups

3. The o-finite class group and the o-centre of G

4. The type I part of V(G,ur)

1.

2.

.)

4.

(i)(iv)

(v)

I

1

4

6

10

t2

t7

23

2B

JJ

33

40

44

55

58

6B

69

80

82

89

CHAPTER IV . GROUPS WITH FINITI DIMENSIONAL IRREDUCIBLE

MULTI PLIER REPRESENTATIONS

1. Preliminaries

2. The main theorems

4, ExamPles

APPENDI X

1. aO and nO

2. ^d^

and CIp^

3. The dual of Q

REFERENCES ,

103

104

110

L20

4.1

4.4

4.5

(i)

SUMMARY

Let G be a locall,v compact abelian group and o a

normallzed Boiel multiplier on G. llle are concerned primarì'ly

with the von Neumann algebra V(G,o) generated by the regular

rrr-Fêprêsêntation p of G defined by

p (g)f (x) = f (g-r x)o(g-1 ,x) '

g e G, almost aì'l X e G, and f e L2(G).

The pair (G,o) is called type I if every

ur-Fêprêsêntation of G is tYPe I.

In Chapter II, we investìgate the structure of (G,r,r) ln

the case where G is abel'ian and (G,r) is type I. In

particular, we can reduce t,he study of such pairs to the case

where G is residually finite. Furthermore, if G is separable

and divisible, then (G'r)'is type I if and only if there

exists a bicont'inuous isomorphism from G to a group of the

for'm H x H^ where H is a closed subgroup of G, which carries

t¡ to a multiplier on H x H^ that is similar to the multip'lier

or' gi ven by

,'((x,i,)(y,x)) = r(y) ,

(x,r)(y,x) e H x H^.

(ii)

Chapter III provides information about the maximal type I

central projection e in V(G,o) in the case where G is a

discrete group. Indeed we have e + 0 if and.only if there

exists a subgroup H of G such that the index [G:H] is finite,

the commutator H'has finite order and o restricted to H is

the trivial multiplier. l^le also show hovl e can be realized as

a convolution operator on t2(e). As a consequence of thjs

result we can prove that for G discrete, (G,r) is type I ifand only if V(G,o) is a type I von Neumann aìgebra and that

this occurs if and only if G has an abelian subgroup A of

finite'index in G such that o restricted to A is a trivial

multjplier. fnis result generaìizes easi'ly to assert that a

locally compact group G with normalized Borel multiplier ur

satisfies V(G,to) Ís type I.O for some natural number k if and

on'ly if G has an open abelian subgroup A of finite index in G

such that the restriction of o to A 'is trivial.

Finally, in Chapter IV these results are generalized to

an arbitrary iocalIy compact group G with Borel multiplier o.

Let e be the maximal type I finite central proiection in

V(G,r¡), then e * 0 if and only if

(i) [G:¡] ( æ

(Íi) a'has compact closure, and

(iii) there exists a finite dimens'ional o-representation

of Â, where a denotes the closed normal subgroup of G

consisting of all those elements whose coniugacy class has

compact closure. Again we can construct e as a convolution

operator on L2(G) and use this to prove that the folìowing

Ir]1j

are equivalent.

(i )

(ii)(iii)

All irreducible ur-representations of G are finite

dimensional.

V(G,o) is type I finite.

The fol l owing propert'ies hol d .

(a) [G:a] < @

(b) ^'

has compact closure.

(c) There exists a finite dimensional

o-representat'ion of ¡.

(d) n{ker n : n is a finite dimens'ional

' (ord i nary) representat'ion of ¡] = { I } .

All the irreduc'ible (ordinary)representations of the central

extension Go are finite d'imensional .

(iv)

Mackey's normal subgroup analysis is used in

conjunction with the above theorem to construct a group G and

multjp'liers or, t e [0,1] such that V(G,cua) 'is type I finite

and has a non-zero type in Part for arb'itrarily large natural

numbens n if t is rational, and is type II, if t is

irrationa'l .

(iv)

This thesis contains no material which has been accepted

for the award of any other degree or diploma in any University.

To the best of my knowledge and belief, the thesis contains no

material previously publ ished or written by any other person'

except where due referenie is made in the text of the thesis.

A.K. Hol zherr

(v)

ACKNOl,.JLEDGEMENTS

I wish to express my appreciation to W. Moran, my

supervisor, for hÍs encouragement and advice during the

time spent on doing research for this thesis, and also

for his support as a friend.

My appreciation 'is extended also to Helen Koennecke and

Chery'le Marshall who did the typing for this thesis.

1

CHAPTER I

PRELIMINARITS

This chapter is introductory in nature. It serves to

define notat'ion, to provide the background, and to introduce

some of the resul ts needed for I ater chapters. For more

detai'led properties of locally compact groups, C*-aìgebras

and von Neumann aìgebras, the reader is referred to [7' 18'

34, 40, 471.

1 Locally compact abelian qroups

Let G be a locally compact abelian group. By this v¡e

vriil always mean a topolog'ical group which is local'ly compact

and To (and hence normal ). We adopt the common usa9e

throushout of ustnn .n:,:oj -fX'åiL. to describe a locallv

compact group which is metr^izaålæ. Group homomorphisms

(isomorphisms etc.) are such strictìy in the algebraic sense

unless it is specifically stated otherwise, A bicontinuous

isomorphism 'is aìso referred to as a topoiogical

isonrorphism. Given two locally compact groups H and K, then

t{ x K denotes theìr direct product (with Cartesian product

topoìogy) and l-l is identifjed (up to topo'logical isoinorph'isrn)

with the subgroup H x ieÌ. Simjìar1y for K" A subgroup ll of

G is called a topologicai djrect summand of G if it is closed

and there ex'ists a closed subgroup K of G such thai G and

H x K are topo'logìca'l1y isornorphìc, and is called a direct

2

sunrnand if the same property hoìds for the discrete topoìogy

on G.

Let G be a locally compact abelian group. The character

group (or duaì) of G, denoted by G^, is a group whose

elements are contjnuous homomorphisms, G +ill'(where Il is the

group of comp'lex numbers of modulus 1 w'ith the induced

topology) and with multiplication defined pointwise. The

sets {x e G^ : x(0) g V}, where C is a conrpact subset of G

and V is a neighbourhood of the identity in'll', is a basis of

neighbourhoods for a locally compact group topology on G^.

If S is a subdet of G, then A[G^,S] = {x e G^ : x(s) = 1 all

s e S] denotes the annihilator of S in G^. The followìng

duatity theorems and structure theorem are well known.

THEOREM 1.1. Let G be a LocaLLy eornpact abeLian gzoup,

then

ø)

øi)

øi¿)

(G^)^ is topologicaLLy isomorphíc aith G.

If S is a subset of G mtd K is the stnalLesi;

cLosed subgroup of G eontaining S, then

AlG, A[G^,S]l = K.

If H and K ave cLosed subgroups of G such that

H s K, then A[K^,H] is topoLogicalLy isomorphíc

uví,th AIG^,Hl/AtG^ ,Kl cnd (K/H)^.

If H ís a eLosed subgroup of G, then 11 is open

(nespectiueLy eompaeÐ íf A[G^'l]l is compact

(respectiueLy open. )

Gtt)

3

PROOF. See [18, 24.8, 24.10 and 23.24).

THE0REM 1.2. A LocaL Ly cornpaet group G is topoLogícaLLy

isomorphie to lRfr x K, uhere K is a LocaLLy compact abeLian

group containíng a eompq,ct open subgroup. L'Lte ínteger n ís an

inuaz,i-ant of G,

For a proof of this result see [18, 24,30].

The following result relates the structure of G to that

of G^.

THEOREM 1.3.

(i)

øi)

ft,ii)

The duaL of a diuisibLe LocaLLy compact abeLían

group is toz,síon free.

The duaL of a conrpact totaLLy dísconnected gt'oz'Lp

is tov,sion.

lf G ís a eompaet group, then the foLLouing ate

equiuaLønt.

k) G is eonrpeted.

ft) G^ ¿s tonsion free.

(e) G is dLuisíbLe,

PROOF. See [18 , 24 .23, 24 .26 , 3.5 and 24.25] .

Finatly, we mentjon some a'lgebra'ic propert'ies of discrete

abel ian groups.

4

THEOREM 1.4.

(i)

øí)

(i¿i)

A diuisibLe subgtoup D of an abeLimt groLtp G ís a

direct sunrnand, Indeed the conrpLement may be

chosen to eontain any subset of G uhích interseets

t:æíoiaLLy uíth D.

If H is a subgroup of G such that the order of

each eLqnent ín H is Less than some fined integet',

anå GIH is torsion free, then H is a direct

sutnnanÅ. of G.

A fínite abeLiqn groLtp is isomorphic uíth a

finite product n Hp, uhere each H, is a cgeLie

groLe of prime poüer order,.

PROOF. See [18, 4.8 and 4.25] and lI2, 27 .51.

2. Concrete C*-alqebras and

von Neumann alqebras

Let H be a Hilbert space with inner product ( ','), and

tet B(H) Ue the algebra of all bounded linear operators on H

and B(H)* the Banach space dual of ts(H). The uniform

topo'logy on B(H) is given by the norm liall (a e B(H)), where

llall - trpiltil<1,8e, llaEll

B(H) is a Banach algebra under this nr¡rm and, with the

adjoint operation a + a* (defined by the rejation

(ô6,n) = (8,ê*n) for all Ë,n e H) as the invojution, B(H) is a

5

C*-algebra. Indeed any norm closed *-subalgebra of B(H) (tf¡at

is, a subalgebra invariant under the action of the involutiotl

*) is a C*-algebra. For a converse we have the following.

üEqßEI_¿¿, ( [40, 1" 16.6] ) . Let; A be a cx-aLsebra,

then thev'e eæists a x-ìsomorphísm from F\ onto a tmífot'mLy

cLosed x-subaLgebra of B(H) for some HiLbert space H.

Given a subset A of B(H) we denote by A' the commutant of

A viz. A' = {a e B(H) : ab = ba for all b e A}, by A" the set

(A')' and by CA the centre A n A' = {a e A : ab = ba all

b e Aì of A.

A *-subalgebra A of B(H) for which A" = A is called a

von Neumann al gebra (on l^l*-al gebra ) acti ng on ll.

The weak operator topoìogy on B(H) is the smallest

topology on B(H) such that for all 6,n e H, the function

B(H) +C : a +(âã,r) is continuous. Each von Neumann algebra

acting on H is automat'icalìy closed with respect to this

topology and conversely, a weakly closed *-subaìgebra of B(H)

containing the identity operator on H is a von l{eumann

al gebra (Sakai [40, 1 .20.3] ) .

Let A be a von Neumann algebra acting on H and let V be

the linear space of all continuous linear functionals on A

with respect to the weak operator topology and denote by A*

the norm closure of V in B(H)*. The o(A,A*) topology on A,

that is the smaliest topology on A such that each element in

6

A*'is continuousr'is called the o-weak (or ultra-weak topoìogy).

A* cons'ists of al I o-!{eak conti nuous I i near ftrnctional s (al so

called normal functionals) and is called the predual of A; its

Banach space duaì can be identifìed (up to isonletric

isomorphism) with A by means of the map A * (Aa)* ' ¿ + Tu,

where fu('t,) = q/(a), â u A and ú e A* ([:2, Theorem 2.4]),

Furthermore, A* is unìque in the sense that no other norm

closed subspace of B(H)* has th'is property ([47, III.3.9] ).

THEOREM 2 2 ¡47, III.3.10l ) . Euery *-isomorphíun from

q.1)on Newnann aLgebra to another is o-ueakLg continuous.

Suppose we have von Neumann alcebras Ar, (i = 1, 2)

acting on Hilbert spaces H' then a *-isomorphism o: A, +A,

is said to be spatiaì if there exists an isometry U of H.,

onto H, such that

(a)=UaU*, acA.

l^le point out at once that *-isomorphisms are not necessarìly

spatial ([40, page 119]).

3. Classi ficati on of von Neumann alqebras

Let A be a von Neumann a'lgebra act'ing on a Hilbert space

H. A self adioint element e of A is called a projection if

e2 = e; a project'ion e is called central 'if ae = ea for all

a e A (tfrat is e e CA = A n A'), and abelian if eAe is an

abelian algebra. Two projections pr Q âFO sajd to be

7

orthogonal if pq = 0.

Let p and q be two project'ions in A. If there exists an

element u of A such that u*u = p and uu* = q, then p'is said

to be equìvalent to q and we denote thìs by p - q. If there

exists a projection q, (< q) equivalent to Pr v,rê write this by

p q or q p. The relation "-" satisfies the conditions of

equ'ivaì ence and the relation rr rr i s refl exive and transitive.

Let p be a proiection in a von Neumann algebra A. p is

said to be finite, ìf for a projection p, in A, P, < P and

Pr - P imply Þ, = P; p'is said to be purely infinite if itdoes not contain any non-zero finite proiection; p is said to

be infinite if it is not finite.

Abel ian projections are finite ([40, 2.2.8)), A]so one

can easi'ly see that along with fìnite projection, all

smaller ones are also finjte. A von Neumann a'lgebra is said

to be finjte (respectively purely infinite etc.) if its

identity is finite (respectively purely infinite etc.).

Let tj, i e J be a fami'ly of mutual'ly orthogonal finjte

central projectiorrs of a von Neumann algebra A and let

z= L zj. Suppose z- p < z. Then pzj a 2.. Hence y=j = =j

(i . J), so that p = z. Hence z is also finite. Hence there

exists a unique maxìma'l finjte central projection zr in A.

Similarly, we can see that there exists a unique maximal

purely infinite central projection z, in A. Set

zz=I-r.,,-23.

I

A is saìd to be semifinite if zs = 0; properly infinite if

zt = 0; properly infinite and sem'ifinite if zL = ,3 = 0,

THEOREM 3.1 ([+0, 2.2.3]), A uon Neumarm aLgebna. A ís

wtíqueLg deeonrposea trtto a div'ect sun of thv'ee aLgebras uhich

æe finil;e, properLa infinite and semifiníte, and pweLy

infínít.e, respeetiueLy .

Finiteness, semi-finiteness can be characterized using

traces. Let A be a von Neumann algebra acting on a Hilbert

space H. A trace on A+ = {a e A : (a6,8) > 0 all E e H} is a

function p deíined on A+, with non-negative finite or infinite

values, possessing the following properties.

(i )

(i i ¡

.p(a + b) = ç(a) + ç(b), a,b . A+,

e(ra) = ¡.ç(a), À e c, a. A+ (with the convention

that o . * æ = 0),

If u is a unitary operator of A, we have

ç(uau-l) = ç(o), a . A*.

(iii)

I is said to be finite if .p(a) . +- for all a e A't; , is

said to be sentifinite if, for every a e At, e(a) is the least

upper bound of the nunlbers,p(b) for the b. A+ such that b < a

and ç(b) f **; ç is said to be faithful if the conditions

a e A* and r(a) = 0, imply that a = 0; and ç is said to t"¡e

normal if it is o-weakly continuous.

THEOREM 3.2 ([40, 2.5.4 and 2.5.7] ) . Let A be a

uon Neumann aLgebra. Ihen à is f'Lnite (r'espectiueLy semi'

9

finite) 1',f ffd. onLy íf fo7 any non-zevo a in Ar, theye eæísts

a norrnaL fínite (r'espectiueLy semifinite) traee ç such that

p(a) * 0- A is properLy infinite k'espectiuely puyeLy

infínite) í,f and only if thev'e is no nor,naL finite(r'espectiueLy semi-finite) trace on Ai ercept for the

ídentícaL zero tt'a.ee,

On a semí-finite uon Neumarm aLgebra, there eæists a

semífinite faithfuL trace.

A von Neumann algebra is said to be type I if every non-

zero central çiroject'ion contains an abel ian proiect'ion; type

II if it is semi-finjte and does not contain any abelian

projection; type III if it is purely infinite. A finite type

I (respectively type II) von Neunlann a'lgebra is said to be

tYpe I, (respectively type IIr), and a properly infinìte

type I (respectively type II) von Neumann aìgebra is said to

be type I- (respectively II*). A central project'ion z in a

von Neumann algebra A is said to be type X (X = I, II etc.)

if zA 'is type X.

THEOREM 3.3. ([+0, 2.2.10]). A uon Neumann aLgebra is

wtiqueLy deeorn¡tosed into a type I, tApe 17 a'nd type I I"I

díz.eet swnnand.

The fol I owi ng d iagram wi I I hel p 'i nterpret these

definitions and facts.

ÏII

æfI I

IIt II@

10.

fi ni te proper'ly i nf i ni te

type I

type II

type III

semi -fi ni te pure'ly i nf i ni te

tVpe I, type I I, tvoe I otvoe IIó type III

4. TypeJ von Neuman al ebras and

pol vnomi al identi ti es

Let A be a von Neumann a'lgebra with centre

C = {a e A : ab = ba all b e A}. A is said to be tYpe In

(n = 1,2, ...) if jt js *-isomorphic to C ø B(H)' where H is

a Hilbert space of dimension n. Note that C ø B(H) is nothing

other than the n x n matrix algebra over C.

THEOREM 4.1, ( [40, 2.3.2]). A type I, uon Neumarvt

algebra ean be decontposed as a direet swn of type In

Ðon Neumarm aLgebras (n e Z).

Using this result, we define a vorì Neumann algebra A to

be type I<k if the type In Part in A is zero whenever n > k.

semi -fi ni tepurel y

i nfi ni te

fi ni te properly infinite

11.

For any natural number k, let SO denote the standard

polynomìal in k non-commuting variab'les

Sk(u,, ..., ao) = f (-l )Y av(r ) uy(r) ... uv(t )

where the sum is taken over all permutations y of {1, ,..' k}

and (-l)Y denotes the signature of the permutation. Let A be

an algebra. l,le say that the ident'itV SO = 0 is satisfied

identically in A (or more brief'ly A satisfies SO) if

sk(A) = {o}

where Sk(A) denotes the set {SO(ar, ...' uk) : all ui e A,

i =tl, ..., k]. Po]ynomial identities are relevant in this

context because of the Amitsur and Levitski Theorem:

THEOREM 4.2 ([39, i.4.1]). For euery eonnwtatíue ring

R, the aLgebna of n x n ma:brì-ces Rn over R sal;ísfies Srn'

we have a partial converse of this theorem which is due

to Kaplanski,

THEOREM 4 3. An n x n matz'ín aLgebra oüer a, fíeLd does

not satísfa SZU fon k < n.

For the rest of th'is Section, we fo'llow Taylor t48l .

THEOREM 4.4. If the uon Neumann aLgebra A is nol; of

type I.n, then th.ev:e eæists a. copA ,f Mn+l fthe

T2,

naturaL nwnbet'.

(n+l)x(n+1) -compLeæ matyiees ) in A.

To see this, note that if A is not of type I.n' then

there exists a set of n + 1 nlutua'l1y orthogonal equivalent

projections in A. As in Smith 144, Lemma 9.31, a copy of

Mn..,.1 can be constructed in A.

THEOREM 4.5. Let A be a uon Neznnartn aLgebra and. n a

Then A satisfies trn Uf anð. onLy if A is of

tApe I=n.

PR00F. ft A satisfies Srn, then by 4.3 and 4.4, A is of

tYpe I=n. Conversely, if A is type I.n, then A is a direct

sum of the algebras A.O, (1 < k < n), where each AO is type

Ik. By 4.2, each AO satisfies Srn. Therefore A sa.tisfies

S2n:

PROP()SiTION 4.6. Suppose B ís a ueakLy dønse subalgebna

of the üon Neumann aLç¡ebra A, then A satisfíes trn Uf B does"

PR00F_. Let a

the I i neari ty of S

1 be a net in A converging weakly to a. By

and because multiplication in A is2n

lveakly continuous, SZn(a1, ar, ..., dZ¡) converges to

SrO(a , dZ, .. ", uzk). Similarly for the other variables.

This proves the result.

5. Representati ons

l.le give a very brief outline of definitjons and required

13.

results. For more details see 17,15, 2,301.

Let A be a Banach *-algebra. A representatjon of A in a

Hilbert space H is a homcmorphism

n:A*B(H)

which is non-degenerate in the sense that n(a)E = 0 all ô e A

impìjes g = 0. Since a *-isomorphism from a Banach *-algebra

to a C*-algebra is norm reduciñg, â representatjon of A is

necessari ly conti nuous .

The (tlil bert) dimension of H is cal I ed the d'imension of

n ônd is denoted by dim n. The kernel ker n of n is the set

{a e A: n(a) = 0}; and n is said to be faithful ifker n = {0}. Two representations n and nrare saici to l¡e

equivalent if there exists a unitary operator U : Hn, + H,,,

such that U-tn(a)U = rr'(a) for all â e A. lnle will not

usually distinguish between a representatio¡t and its

equiva'lence class"

Given a set of representations r , j . J, of A, de'f,inejtheir direct sutìt,o¡eJ nj by letting A act on the Hilbert

space tj.J

and each Hj

H, coordinate-wise. Conversely, if H = *j.¡ flj

invariant under the action of A (via n), then

n = tj.J nÏ where n, (i . J) is the restriction of n to llr.

Suppose we have two representations Tr and n' of the

Banach *-algebra A. Their tensor product is the

14.

representation n @ n' whose associated Hilbert space is

H o H , and js defined byTt 1Í'

(n e n')(a)(g * E') = tt(a)E t tt'(a)6' e

E . Hn, E e Hn,, a e A.

Denote by V(n) = {n(a) : a € fi1" fhe von Neumann algebra

generated by the operators n(a), â e A. V(") is iust the

weak closure in B(Hrr) of the complex linear span of the

operatorstr(a),ôeA.

n is callecl a factor (or primary) repnesentation if the

centre of V(n) consists of scalar multipìes of the identity

in V(n)(that is CV(") = 0.I).

PROPOSITION 5" ]. ( [40, 1.21.9] ) . Let n be a repnesenta-

tion of the Banach *-aLgebra A. The foLLouing are

equíuaLent.

(i)(í.i)

V(n) = B(Hr).

If H is a eLosed subspoce of Hn uhích is

inuavíant, that is n(a)6 e H aLL Ë e H, ô e A,

thenH = {0} oz'H=Hn.

If either of these equivalent conditions are satisfied,

then n is called ireducible.

Suppose we have an irreducible representation n and an

15

operator T such that n(a)l = Tn(a) all a e A, then by 5.1' T

commutes with all the proiections in B(Hn) and thus T 's

[15, IV.3.9].+-ísæ+i+ens+en++. This js called Schur's

Lemma.

co*L..-t

We will need a v,/ay of passing from Banach *-algebras to

C*-algebras. Suppose n is a representation of the Banach

*-aìgebra A and let a e A. Since n is norm reducing,

iln(a)il < llall , thus the supremum of iln(a)il as n runs through

all the irreduc'ible representations of A is a well defined

number which we denote by llall '. Let I be the set of â e A

such that llxll ' = 0, which is a closed self-adjoint two-sideci

ideal of A. The map a + llall ' defines a norm on the quot'ient

AlI. Endowed wjth this norm, A/I sat'isfies all the c*-algebra

axioms except that A/I is not complete in general. The

completion of AlI is a C*-algebra called the enveloping

C*-algebra of A and is denoted by C*(A). The canonical map of

A into B is a norm-reducing *-homomorphism whose image is

dense in B. when A is a c*-a'lgebra, we can identify A with

c* (A) .

THEOREM 5.2, (17 , 2.7.41). Let A be a Banach *-aLgebra

uith m approxùnate ídentLta artd r the carPnícaL map of A

into C*(A).

rf n is q iaepresentation of A' thev'e ís eæactLy

one repv'esentation p of C* (A) such that

(ì.)

IT = P o T.

ftÐ The møp t + 0 is a biiectíon of the set of

16.

('1 1,1, )

Gu)

representations of A onto the set of

repz,es entations of C* (A ) .

r is ívu'e&,¿cibLe ¿f qn¿ onLy if p is irueducibLe.

V(n) = V(p).

This result shows that the maiority of questions

concerning representations of Banach *'-algebras with an

approx'imate identity, it is enough to dea'l only with the

C*-algebra case.

The following result ensures an adequate supply of

representati ons .

THEOREM 5.3 $+l , I.9.231). Lel; A be a cx-aLgebz'a arú.

Let a ín à be non-zeyo. Ih.en they,e erí,sts an í,zryeducibLe

z,epnesentation r of A such that n(a) * 0.

A representation n of a C*-algebra A is said to be type

I (respectively finite etc.) if the von Neumann algebra V(n)

is type I (respective'ly finite etc.). A is called type I if

V(n) is type I for all representations n of A.

Let A be a c*-algebra. The set of equiva'lence classes of

irreducib'le represenbations of A denoted by A^ is called the

dual of A. The dual A^ is given the hull-kernel topology

which is defined as follows. A subset F of A^ has closure

¡- = {n e A^ : ker T f nr'.¡ ker n'}

17.

'Type l-ness' of a separable C*-algebra (tfrat is one that

contains a countable dense subset) can be characterized using

this topology. This is a deep theorem due to Glimm.

THEOREM 5.4 [7 , 9. ]. and 9.5.21 ) . Let A be a C*-aLgebra'

then A is type I íf anÅ onLy íf euera factor representation is

type I. Îf in oÅ&Ltion A is sepæabLe, then A is type I if

and onLy if A^ ís a I o topoLogieaL spaee,

6. Mul ti pl i ers and mul ti I ier representat'ions

Let G be'a locally compact group with identity ejement e"

A multiptier (or cocycle or factor set) on G is a Borel

measurable funct'ion o: G x G +'T'(the group of complex

numbers of modultls 1, with the induced topology) which

sati sfi es

r(g,h)r(gh,k) = o(g,hk)o(h,k)

ur(e,g) = ó(g,e) = 1

g,h,k e G,

g e G.

Two multipliers o, and t,l2 are similôr rrr, * az, if there

exists a Borel measurable function Y : G +'lf such that

*r(s,h) = Y(g)v(h)r(3h)-t 'r(s,h)g,h e G.

A muttipt'ier which is similar to f (the constant function on

G x G) is said to be trivial. Every multiplier is similar to

a normalized multiplier, that is one vrhich satisfies the

addi tional property

18.

,(g,g-I) = 1, g € G.

Indeed, if o is an arbitrary multiplier, then the multipljer

9,h + *(gh, (gh)-1) ,2,(g,g-I )

-r2 ,(h,h-1 )-4 ,(g, h)

is normalized. The square root here is taken in a Borel

measurabl e fashion.

If t¡ is normalized, tlten using the cocycle identity' we

have

r(g-1,h-1 ) =,¡(g-r,h-1) u,(g-1h-I,hg)

= ,(g-t,g) r(h-t,hg)

= o(h-t,hg),

and

o(h,g) r(g-1,h-1) = r(h,g),,,(h-I'hg)

= t¡(hh-1,9)r(h-1,h)

= 1,

that is t¡(h,9)-i = r(g-trh-1), al'l h,g e G.

Each normalized multiplier on G defines an extension G

of 'lt by G. It is the set "11'x G provided with the

mul ti pl ication

UJ

(s,g)(t,fr¡ = (stro(g,h),gh)

19.

(and then because æ is normaljzed, (s,g)-r = (s-l,g-1)).and a

topoìogy in which a basis of neighbourhoods of the identìty

'is composed of the sets AA-r, where A is a set of finite

positive measure for the product of ríght Haar measures on'll

and G. This is the topo'logy defined by Weil cn groups w'ith

an invariant measure (Weil [51, Appendix 1])" It is easy to

check that this topoìogy jnduces on 'F (indentified with the

ôentral subgroup'l[ x {e} of Go) its original topology and

makes Go into a topological group extension of ]t by G. Since

both'll and G are locally compact, so is G'. Go is unique'ly

determined, to within topologica'l isomorphism, by the

similarity class of o. In fact 'if o, - u¿, say

,r(h,9) = v(h)v(g)v(hg)-1 ,r(h,9), then the map

o : (t,g) -, (tv(g),g) is an isomorphism of Gtt onto G'2. By

Fubini's theorem, O iS measure preserving and it follows from

the definition of the topology of these groups that o is

biconti nuous.

Kl eppner (126 , Lemma 2I ) has g'i ven the fol'l owi ng

alternative definition of the topology on G'.

PROPOS ITI0N 6.1 , (1,26, Lemma 2l ) . The set;s

(U ,. f )(U * F)-1, uhev,e U runs through a basís of

neighbouthoods of I ìn T aná t through tlte sets of posi.tioe

measu,re in G, forTn a basis fon the neighbounhoods of (I,e) in

^ (¡¡t¡.

Let H be a closed subgroup of G. Then to restricted to

H x H is a multiplier on H which we also denote by ,, and Ho

24.

is algebraical]y isomorphic to a subgroup of Go. Moreover, we

have as an immediate consequence of Proposition 6"1, the

fol I owj ng resul t.

PROP0SITION 6.2 ([3, Lemma 1.1]). Let H be a cLosed

subgroup of G" Then the ineLusion map H' * G ^ í," o

bicontínuous isomoz'phism of H'uith a eLosed, subgz'oup of G^.

Let G be a locally compact group and o a Bore'l

mul ti pl i er on G. The twi sted L l-al gebra L t (G, o; i s the space

LI(C) of complex-valued integrabìe functjons with

mul tì pl i cat'i on def i ned bY

f * h(x) = f(g)h(g-1x) (g,s-rx)dgG

and with a *-operation defined bY

f* (x) = f (x-i )- ¡(x-1)

where - denr¡tes compìex coniugation and ¡ is the nlodular

function. One verÍfies at once that LI(G,to) ìs a Banach

*-algebra possessing an approx'imate identity, which is

determined up to isomorphic *-'isomorphism by the equivalence

cl ass of co. In general Lt (G, r.,r) i s not a C*-al gebr a (17 ,

13.3.61 ) . However Theorem 5 .2 app'l i es . The C* *compl eti on of

Ll(G,ur¡ is called the tr^¡isted group C*-algebra of (G,or) and

is denoted by C*(G,r). If t¡ is identically I' ihen u¡e

delete all mention of it so that C*(G,r) becomes C*(G) which

is called the group C*-algebra of G.

21

Let G be a locally compact group and o a Borel

multipl'ier on G. The construction of C*(G,o) is useful

because its representations correspond to multiplier

representations of G. A mult'iplier representation (or

6-representation) in a Hilbert space Hn'is a map of G into the

space of r¡nitary operators U(Hr) such that

g -+ n(g)E is measurable, E . Hn,

n(g)n(h) = ,(g,yh)n(gh)' 9,h < G.

The concepts oî 'd'imension', 'equivalence', 'direct sum',

'V(n)', 'irreducibility', and 'factor' that we defined for

representations of algebras apply to this situation equally as

well, they are independent of the object being represettted.

For details see Gaal [15, IV.1, page 145]. [l|e let the reader

make the obvious definitions. However if nt is an ot'

representation of G (i = 1, 2), then *r a n2_ (¿etined in a

manner similar to that used for representations of algebras)

i s an orrurr-FêpFesentat'ion, where uoro, i s the mul ti pi i er

(g,lh) * u,r(g,fh)rz(g,h),g,h c G. The set of equivalence

c'lasses of irreducible 0r-representations ìs denoted by

(G,r)^. An o-representation where the mul tipl ier o is

i denti cal 'ly 1 i s cal I ed an ordi nary representati on or s imp'ly a

representation of G"

Mackey has shown that n * no where

,,0(t,g) = tnn(x), (t,g) . Gs, n e z

DC

is a bijection between the set of equivalence classes of r[-representations of G ancl the set of classes of (ord'inary)

representations n0 of Go such that n0(t,g) = thrr0(1,g), al1

(t,g) . G' (see [26, Corolìary to Theorem 1]).

The following theorem establishes the desired connection.

THEOREM 6.3 (17, 13.3.5l ). Let G be a LocaLLy contpaet

group and u a BoreL rruLtipLiez' on G. Fon each u-representation

r of G, put

n'(f¡ = "(g)f(g)dg,G

f e l-l(G,c,r¡, then ' : îr -> îr' is a bíjeetion betueen

u=Tepvesentations of G and. nepresentations of the Banach

*-aLg,zbra LI (G,t'r ¡.

This together with Theoren 5.2 sets up a one-to-one

comespondence between ür-representat'ions of G and the

representations of C*(G,r). As the notation already

suggests, this correspondence preserves d'imension,

irreducibil ity, equìvalence, V(n), direct sum and primaryness

etc. We use it to transfer the topology on g*(G,o)^ to

(G,r)^. One can show that for G abelian and t¡ trivial, this

topology on (G,r)^ coincides with that given for abelian

groups in Section 1. Note that the above remark together

with Theorem 5.3 also shows that there exist imeducible

o.-representations of G for every multip'lien o. l,Je say that

the pair (G,r) is type I (or G is r¡-type I) if C*(G,r) is

o')LJ.

type I.

7. Moore qroups and the reqular representation

Let G be a locally compact group. ble adopt the fol'lowing

notation consistently throughout. Let H be a subgroup of G,

then [G:H] denotes the index of H in G, H- the closure of H

and H' the commutator subgroup of H - the subgroup of G

generated by the elements {ghg-t¡-t : 9,h e G}. If dim n is

finite for all r e G^, then G is called a Moore group (after

C.C. Moore who characterized these groups 'in t31l ). For any

local'ly compait group G, the von Neumann kernel is the closed

normal subgroup Go of G given by

G =O{ker n : ri < G^ and dim t < -}0

THEOREM 7.1. Let G be a LocaLLy cornpaet gnoup and H a

cLosed subgr.oup of finite indeæ in G, then Ho = G0.

PR00F. Suppose X e Ho and n is a finite dimensionai

representation of G, t.hen n restricted to H is a finite

dimensional represenbation of H, So rr(x) = I. It follows

that Ho s Go. Since H/H' is maximalìy almost period'ic and

lG/Ho : 1'llHoJ is finite, by Heyer [19, Satz 7.2.2], G/11o is

also maximal'ly almost period'ic, thus Ho = n{ker n : n is a

finite dimensional representation of G lifted fnom G/HoÌ z Go.

Let G be a loca1ly compact group. We denote bV Gtt the

topological finite class group of G, that is the closed

'A

subgroup of G consisting of the elements x in G such that the

set {g-lxg : g e G} has compact closure. If G = GFC, then G

is called a (topological ) FC group.

These concepts alone allow for a characterization of

Moore groups.

THEOREM 7.2 (Robertson [38, Theorem 1]). Let G be a

LocaLLy eontpact gï'oup. 'Ihen G is a Moore group if d'rld' onLy if

G satisfies the foLloníng properties.

(i)

(ii)('L1,1, )

íe:Gpçl<æ,(Grc ) '- is contpact, and

Go = {e}.

This theorem is due to Robertson. 0ther proofs can be

found in Kaniuth [23, page 233] and Poguntke [35' Satz 3.4].

The proof is far from trivial and we omit it.

The statement of 7.2 can be changed slightly' without

much difficulty, to give the following.

PROPoSrTrON 7.3 ( t38l ) . The LocaLLy conr¡>act gz'oup G ís

Moore if and. onLy if there eæists a eLosed sttbgt'oup K of G

sJrch t?tnt [G:K] ( -, Kl' is eornpaet md Ko = {e}.

Kaniuth (t231) has shown that an SIN group, that is a

locally compact group which has basis of neìghbourhoods at

the identity consisting of sets which are invariant under the

25.

G-action of conjugation, is a Moore group if and only'if it is

type I.

Moore groups can also be characterized using the regular

representation. Let G be a locally compact group. The left

reguìar representation p of G in L2(G) is defined by

(P (g)f )(x) = f (g-rx) '

X,9 € G, f e L2(G). It is clear the p is indeed a

representation. Similar'ly we define the right regular

representatiorí r

(r(g)f)(x) = f(xg)¡(x) h

9,x e G, f e L2(G), where in this context, ¡ denotes the

modular functjon. Denote by v(G) tne von Neumann algebra

V(p) generated by the operators p(g), g e G, and by V'(G) the

von Neumann algebra generated by the operators r(g)' g e G.

The folìowing theorem relating V(G) to V'(G) is much

deeper than its statement indicates.

THEOREM 7.4 ([46, Theorem 3]). Let G be a LoeaLLy

compaet gaoup. Adapt the abotse notation, thett Y '(G) = V(G)'

(on equiuaLentLy V'(G)' = V(G)).

PROPOSITION 7.5 (Taylor [48, Proposition 4.i]), Let G

t

be a LoeaLLy conpaei gz,oup. Tnen V (G) ¿s a finite uon Neunartn

26.

aLgebra if anã onLy íf G is a SIN group.

Thus if G is a SIN group which is type I, then V(G) is

type I, (tf¡at is both type I and finite. See Section 3.)

Indeed the following is true.

THEOREM 7.6 (Kaniuth 123, Satz 3l). A LocaLLg cotnpact

group G is a tr4oore gï'oup if anÅ. onLy if V(G) ¿s tUpe If .

This was first provecl by Kaniuth in [23]. An alternatjve

proof appears 'in Taylor [48, Coroìlary 1 to Theorem 4]. We

can combine al'l these results in one sjngle statement.

THE0REM 7.7, (t23, 3Bl ). Let G be a T.oeaLLy eornpact

group. The foLLouing statements ate equíuaLent.

ø)

(ií.)

('t 1,'t, )

G is a Moone group.

V(c) ¿s type lr.The foLLouing propev'ties are satisfíed.

lc: GFCI ( æ

(og) is cornpact, and

Go = {e}.

Let G be a locally compact group and e the maximal type

I, Projection in V(G). l^le will be interested jn conditions on

G such that e r 0. Kaniuth has given such conditions.

THEORIM 7.8 (Kaniuth [23, Satz 2] ). The ma,æimaL tgpe It

27.

centna.L projection ìn U (G) is non-zero if an'd only if the

foLLouÌng condítions lrcLd.

(í,)

øi)tG,Grcl < @

(GrC)'- is conrpact.

G is fupe f.

V(G) ¿s type I (or equiuaLentLy type If).

Ihe centre of GFC, that is ig . G : gh = hg aLL

h e Gpç \ has finite ivt¡7eæ ín G.

G is a Moore group.

An alternative proof of th'is theorem appears 'in Taylor

[a8] ; he al so proved the fol I ow'ing.

THEOREM 7 .9, (Tay'lor [48, Theorem 4 and Corol I ary 4] ) .

Suppose the manimaL type I, centi"aT' proiection e in V (G) í.s

non-zero, thevi the Don Newnøtn ketweL Go is corupaet anÅ eV(G)

is spatiaLLy isomozphic úo V(G/G').

As we pointed out earlier, for SIN groups, V(G) is type

I if and only if it is type If. In particular this is true of

discrete groups. Both Kaniuth and Thoma have given

characterizat'ions of type I discrete groups. We state both of

these in the following resuJt.

THEORTM 7.10 (Thoma [49], Kanjuth 1221, Smith t45l).

Let G be a. diserete gz'ottp, then the foLLoaing az'e equittaLent.

ø)

ftì.)

ø¿i)

ftu)

28.

The equivalence of (i) and (iii) is due to Thoma and that

between (i'i ) and (iii ) is due to Kaniuth. An alternative

proof of '(ii ) ìs equivalent to (ì j'i )' is g'iven by Smith t451 .

The main difficulty of this theorem lies in constructìng an

abelian subgroup of finite index in G. Different ways of

dojng th'is (us'ing different but equ'!valent hypotheses) can be

found in Isaacs and Passman [21] and Schl ichting 1421.

Much of this thesis cons'ists of results genera'lizing the

theorems of this section to locally compact groups with non-

trivial multip]iers, and we do this mostly using the methods

developed in Smith t45l and Taylor t481.

B. Induced repre sentati ons, Mackey's constructi on

Let G be a separable ìocally compact group, K â closed

subgroup, o â Borel multiplier on G and n âh o-Fêpresentatìon

of K, The induced representat'ion t13 ts an ür-representation

of G defined on a Hilbert space ll. Following Auslander l2l,we let a be the measurable function cr (g,X) = d(g.u )/du 'g € G, x e G/K, (for more details see a'lso Mackay Í271 and

Blattner t4l). Define H to be the space of measurable

functions f from G to H,, (that is,9 +(f(g),8> is measurable

for each E € Hn) such that f(gk) = n(k)-Ir(g) and

lrl e t2(G/ K,u). The last condjtjon needs a note of

explanation; observe that lf (gk)l = lt(g)l s'ince n is unitary,

so that lfl is realìy a scalar function on G/K. Then define

t'fflts)r) (s) = f(g-1 s)o (g,h(t))'%(s"r,5)-r t

29.

where h is the canonical map from G to G/K. One verifies

ihat î,ff is a unitary operator ancl that g *.lltg) is an ur-

representatìon of G. One also verifies that nffl, up to

unitary equivalence does not depend on the choice of u as the

notation already suggests. The notion of induced

representatjon is compatible with tak'ing central extensions,

as indicated in the following result. If n is an

o-representation of a group G, denote by tto the corresponding

(ordinary) representation of the group extension Go.

PROPOSITION 8.1 ([3, Lemma 1.2]). Let Kbe a eLosed

subgroup of a'separabLe LoeaLLy eonrpact gnoup G uith

norrnaLized BoreL muLtipLiey u. t¡ is an u-Yepresentation of

K, tt"n GIl)o od "|ii

ore equiuaLent.

PR00F. It is quickly checked that the map f + f', where. ^û)

f ,(x) = f (1,x), for all f in the Hilbert space of ,,OJu,., setslHt-

up the desired equivalence.

THEOREM B 2 (Mackey [27, Theorem 4.1] ). Let H and K be

cLosed subgnoups of a separabLe LocaLLy compczct gtoup G uith

Boz,eL rm,fltipLien u, srch that H E K. If r is an

u-repr.eserLtation of H, ttnn ,!f; is equiuaLent to t"1[11fl.

THEOREM B. 3. (Mackey [27, Theorem ].0.11). Let G be a

sepaz'abLe LocaLLy eornpact group uith BoreL rrtLltípLier u. If

H ì,s a eLosed subgrot'tp and n ¡, i . J a eoLLeel;ion of

6-r.epv,esentatíons of H, then (e1.-lnj)ff; ;" equiuaLent to

*r., ("rîff)'

3C

Let G be a separable local]y compact group with rnultipìier

o, ônd H a closed normal subgroup such that (H,r) is type I

(see Section 6). If n e (H,r)^ and 9 e G, then n9 will be the

o-representation of H defined bY

ng(h) = r,(9-1hg)r-1(h,g)r(g,g-1hg), h e H.

It is easy to check that the action 9 * (n -, n9) of G on

(H,r)^ satisfies n9h = (nh)9, ail h,g e G. The set {g e G:

n9 it equivalent to n] is a closed subgroup of G cajled the

stabilizer of n and is denoted bY Kn. An orbit is a subset

of (H,r)^ of dhe form {n9 : g € G} for some irreducible

r,l-representation n of H.

The dual (H,r)^ has a natural Borel structure as

defined in Auslander [2]. Fell [11., page 95] has observed

that this Borel structure is iust that generated by the

topo'logy of (H,r)^.

Now each printary rrr-Fêpl"êsêntatiorì n of G determines a

projectìon valued measurê u',, oh (ll,o)^ which is unique up to

equivalence (two measures being equivalent if they ha.ve the

same null sets), whose values are projectìons on the Hilbert

space of the representation n, and which'is a quas'i-orbit in

the sense that

(i ) un(g.A) = ur,(A) for alt Borel sets A in (H,ur)^

andgeGrand

if A is an 'invariant Borel set (tfiat is g.A = A(ii )

31.

for all g u G) then either A or its complement

has un-measure zero,

(tnat is, Þn is G-invariant and ergodic). For deta'ils see

Mackey [28, 30] and Auslander l?1, The measut. 'rrn is said to

be transit'ive if it is concentrated on an orbit o (that is

u,r((H,.)^ - o) = o). '

THE0REM 8.4, (Effros [8, Theorem 2.6]). Let G be a

sepa.z,abLe LoeaLLy cornpact grory and w a BoreL rm'út'LpLier on G-

Let H be a eLosed norrnaL suhgroup sueh th.at (H,r) ís type I.

If 1;lrc G-orbíís ¿n (H,r)^ are LocaLLy cLosed (a set is

LocaLLy cLosed ¿f it ís the intez'section of an open anÅ.

cLosed set), then v*is transitiue for each pnimary

u-r.epresentation x of G. ConuerseLA íf (H,t)^ has 17þ non-

transitiue ergodic meq.sLlres, then the G orbits in (l'lro)n are

LocaLLy cLosed.

THEOREM 8.5 (Mackey [28, Theorem 8.1]). Let G" u and H

be as ín Theorem 8.4. Let 0 be at¿ orbit ín (H,r)^ and r an

eLement in o uíth stabiLize, Kn. Then the mapping À * ÀffTT

sets up a one-to-one corvesponÃ.ence betueen the 'pr"Lmary

$-r.epresentati.ons of Kn uhose restriction to H is a ntuLtipLe

of r" and the pr'ìmar'y u-?ep?esentations x of G such that v*is

concentrated on a. Fu.r'therTnore, the two Don Natnnqnn aLgebras

Y(x) and. V(ÀTl ) oon *-ísornorphie"1t

THTOREM 8.6 (Mackey [28, Theorem 8.2]). I'et G, u and. H

be as in Theorenr 8.4. If n is an irreducíbLe u-?epresentation

at

of H, then thev,e eæì.sts a muLtípLierc of Kn/|1 and a "ctu'

representation r' of Kr uhere r' ís the Lifting of r to Krr,

srch tLuzb n'(h) = n(h) aLL h e H.

TIIEOREM 8 7 (Mackey [28, Theorem 8.3]). Let G, o, H, rr

Kn, r' and. r be as in Theorem 8.6. Eoz' each r-r-nepresentation

ì, of KnlH, denote bA \' the (r')-L-rnpresentation of Rn

obtained by conrposing )' utLth the emtonícaL m6P Kn -t Kn/H. Then

À -> À' e fi' sets up a one-to-one covrespoluJerrce (eqttíuaLent

represenl;qtionß being identified) betueen the set of prLrnat'y

,-L-nepz,esental;ions of Kn/H ar¡Å. the set of pr"imary

u-:repvesentations of Kn zohich v'edtrce on H 1;o a truLtípLe of n.

Ew,bherrnoz,e, the tuo Þon Neumarm aLgebyas V(À) an¿ U (x' ø n')

ave *-isontonphic.

l,'le can summarize Theorems 8.4 to 8.7 as follows. Let G,

rr, H, r, Krr, n' and t be as in Theorem 8.6. Then

f * (f' ø rr')J G

KTT

sets up a one-to-one correspondence between t.he primary

t-I-representations of K.n/H and the primaF) trr-Fêpresentations

x of G such that u* is concentrated on the orbit contain'ing r.

The two von Neumann algebras V(¡,) and V((r'* '')1f") are

*-isomorphic. Furthermore, if the G-orbits in (H,r)^ are

locally closed, then as n vôries through (H,o)^, the above

construction yields all the primary o-representations of G.

33.

CHAPTER I I

ONT HE STRUCTURE OF O-TYPE I

LOCALLY COMPACT ABELIAN GROUPS

Let o be a normaljzed mult'iplier on a localiy compact

abelian group G. Baggett and Kleppner I3l have given a

useful criterion for deciding when (G,r) is type I. Holvever

not much is known about the structure of type I pairs (G,r).

In this chapter we investigate the structure of ìoca'l'ly

compact abel ian groups that admi t o-type I mul ti p],r'ers. In

particular, a comp'lete structure theory for a certain class

of such groups is given. (See Theorem 4.5, 5.11 and

Corollary 5.12.)

1 Notation and elementary facts

Throughout this chapter, all groups are local1,rr

compact and abelian (tnis includes discrete groups), and all

multipliers are normalized and Borel measurable, unless

otherwise sta+,ed. [,le adopt the notation of Chapter I

Section 1.

Let o be a normalized multipìier on the locally compact

abelian group G. Denote by õ the ant'isymmetrized form of o,

that is

õ(k,g) = r(t,g)/r(g,k)

?¿,

k,g e G.

PROPOSITION 1.1. ([25, Propositjon 1.1, Lemma 7"1 and

Lemma 7 .21). Let G be a LocaLLy eompaet abeLian group and u a

norrna1,ized rm'LLtipLier on G, I,/e haue

G)

Gi)

õ : G x G '>Tl'is a eontinuotæ bicLnraeter.

g * õ('rg) is a continuotæ homomorphism from G to

G

(ì.ii) õ(k,g) = 1 aLL k,9 e G if qnd onLy íf u is

tríuíaL,

The map g * ñ(.,g) is also denoted by the s¡nrbol õ. No

confusion should arise from this ambigu'ity.

PROOF. (i) The bilinearity of õ follows from the

equati on

õ(gh,k) ='(gh,k)r(k,gh)"I

= r(gh, k)r(g,h)r(g,ir)-tr1k,gh)-t

= r(g, hk)o(h,k)r(kg, h)-1r(k,g )-1

= r(g,hk)o(h,k)o(g,k)r(g, k)-1,(gk, h)-1r(k,g )-r

= o(g, hk)o (h, k)o(g, k)r(g, kh) -rr(k, h) -1'(k, g )-r

= õ(g,k)õ(h,k)'

g,h,k e G. Clearly õ(k,g) is a measurable character in k for

fixed g e G. It follows that õ(k,g) is continuous in k for

f ixed g (t18 , 22.191 ). Sirnilarly o(k,g) is continuous in g

for fixed k. By [26, Corollary to Lemma 1]' õ is

35.

continuous as a function of two variables at the identity in

G x G. Using this, we show that õ is cont,inuous. Let (g'h)

be an arbitrary point of G x G and let V be a neighbourhood of

1 in'll'. Let V, be a neighbourhood of 1 ìn T such that Vl s V.

Because õ is continuous at the identity in G x G, t,here exists

a neighbourhood U of the identity in G such that õ(U,U) s Vr.

The neighbourhood can be chosen so that õ(U'g) s V, and

õ(t<,u) s vr. Now õ(ku,su) = õ(k,g)õ(k,u)õ(u,g)õ(u,u)

õ(k,g)Vl s õ(k,g)V and õ is continuous at (k,g).

(ii) Since õ(",g) is a measurable function of G, as in

(i), it is continuous. That õ : G + G^ is a homomorphism

follows from the bil inearity of õ. Fclr the continuìty, ute

observe that the sets p = {X e G^ : X(C) s V}, where C is a

compact subset of G and V a neighbourhood of the identity in

11, is a basis for the neighbourhoods at the identity in G^.

Since õ : G x G +'ìl is continuous, if go . W = {g e G :

õ(g) e P] is fixecl and k e C, there ex'ists an open set U x U'

containing (k,go) such that õ(U,U') s V. But C being compact

can be covered by a f Ínite fam'ily Ur' ..., Un of such sets U,

and if U'= ni=l,...,n U.l is the jntersect'ion of the

corresponding U' sets, we have õ(C'U') s V. Thus if go e W'

there exists an open set U'such that g0 € U's l'1, proving

that W is open and õ continuous.

(iii) Let n be an irreducible ur-representation of G

(such representat.ions exist - see remark follo¡jng Theorem

I.6.3). If õ = 1, then n(g)n(h) = n(h)"(g) for" aì1 g,h e G,

thus by the remark following I.5.1, n is one dimensional and

to is trivial .

36

A normalized multiplier o on a locally compact abelian

group G is called non-degenerate (or totally skew) if õ is an

inject'ion. Given any subset S of G, v'/e denote by Sur the

subgroup So = {g e G: o(s,g) = 1 all s e S}. Since So is

the intersection nr.r{S}o of closed sets, it must be closed.

S is called isotropic if Sur = S and max'imal ìsotropic ifSo = S. An application of Zorn's lemma (see Hannabus II7,1.61) ensures that each ìsotropic set is contained in a

maximal isotrop'ic subgroup. The operation on sets S + Sr'r is

inclusion reversing and (uSo) = n(Sour),

So= {g e G : õ(s,g) = 1 all s e S}

= {g e G : õ(g) e A¡G^,SI}

= (õ)-r (A tG^,Sl ),

thus ñ(So) = AIG^,sl n õ(e ) and

AtG,õ(S)l = {g e G : x(g) = 1 all x e õ(S)}

= {g e G: r(s,g) = 1all s e S}

=So.

The closure of õ(G) is a cJosed subgroup in G^, thus by

r.i.1(ii),

õ(G)- = AIG^,4[G,õ(G)]l = AIG^,Grl .

S . G,nyFor a

Since Gur = ker õ, if r¡ is non-degenerate (tfrat is Gt¡ = ieÌ),

37.

then the range of õ is dense in G^.

PROPOSITION 1 2 . I'et u be a non-degenerate mul'típLiez' on

G. Then it is an open map if, and onLy i,f ß is a bicontinuous

ísomorphísm, Moreoùer,, íf G is separable anå. l;he range of ã

ís cLosed, then ã ís a bí¿ontínuous isomozphism.

PR00F. If õ is an open map, then õ(G) is open and

closed, hence õ'is onto. If G is separable and õ(G) = G^, then

by [18, 5.291, õ is open.

The condition that õ is a bicontinuous-isomorph'ism

eradicates a great dea'l of pathology and forces some order

onto the structure of the group G (for instance'it must be

self dua'l ). It is not surprising then that it is equivalent

to o being non-degenerate and type I.

THEOREM 1.3, (Baggett and Kleppner [3, Theorem 3.2]).

Let u be a non-degenerate rm'LLtipL'Ler on the LocaLLy cornpact

abeLian gnoup G, then (G,r) ís type I if, anå. onLy if

õ : G + G^ : g + õ(",g) i,s a topo\ogicaL isomorphism.

PR00F. lnle give an outl ine of the proof of thìs Theorem

for G separable. For more debails and for a proof in the

case where G is not separable, see llannabuss t17l and Baggett

and Kl eppner t3l .

Let G be separabìe. By 1.3 it is enough to prove that

(G,o) is type I if and only if õ (C) is cìosed. Let H be a

38.

maximal isotropic subgroup of G. Since o restricted to H is

trivial (1.1 (iii)), (H,r)^ is isomorphic to the abeljan group

dual H^. Suppose õ is a topologicaì 'isomorphism. Then the

G-orbit of an element in H^ is of the form {õ(.,g) : g e G}

which is closed by assumption. Thus by Theorem I.8.4, Mackey's

construction with H as the closed normal subgroup gives all

the factor representations of G. Because u is non-degenerate,

the stabiljzer of the orbit {õ(.,g) : g € G} is H, thus the

o-representations obtained using Mackey's construction are all

type I.

Conversely suppose (G,r) is type I. Again by the non-

degeneracy of o, the stabilizer of each element in H^ is iust

H. In particular the action of G on (H,.)^ is essentially

free, and under these condÍtions, the projective version of

Auslander 1,2, II, Proposit'ion 3.11 asserts that there are no

ergodic measures on (H,r)^ which are not transitive. Thus

by Theorem I.8.4, the G-orb'its in (l-t,ur)^ are local'ly c]osed.

We decluce that the subgroup {õ(.,g) : g e G} of G^ is loca]ly

closed. But we know that its closure is G^, so {õ(',9) :

9 u GÌ is an open subgroup of G^. Open subgroups are also

closed, so it must be all of G^.

PROPOSITION 1.4. Suppose Gto g S for some subset S of G

and i:(G) ís cLosed (in partí,cuLan' tLtis is trwe df u is rnn-

degenenate arú- type I)" then

õ(Sr) = A[G^,S], anã

(St¡)o = K ,

39

uhere K is the smaLLest cLosed subgz'oup containing S.

PRC0F. From our earJier remarks, õ(G) = A[G^,Gur] =

A[G^,S], so

õ(So) = A[G^,S] n A[G^,Gur] = A[G^,S], anC

(So)o = A[G,õ(So)] = A[G,AlG^,S]l = (.

We collect this result together with some other

elementary facts as a proposition.

PR0POSITION 1.5. Let u be non-degenenate and l;gpe I on

the LocaLLy conrpact and. abeLian group G. Suppose K and L are

cLosed subgroups of G such that K c L" then

(í. )

fti)

('t'1'1, )

Øu)

(Ko)o = K, õ(Kur) = A[G^,K] an¿ Kt¡ = AIG,õ(K)1.

K is eonrpact k,espeetíueLy open) if, and onLy if

Ku ís opøn k'espeetiueLy cornpact).

(L/K)^ is topoLogieaLLy isomorphie uíth Kr¡/Lr,r.

rf So is a coLLeetion of subsets in G" then

(nSo)ur = (u(Sour))-.

PR00F. (i) follows from 1.4 and earìier remarks, and

(ii) follows from (i),1.3 and I.1.1.(iv). To see (iij)' use

I.1.1. (i i i ) and observe that

Kur/Lr,r: õ(Kr¡)/õ(Lur) = A[G^,K]/AtG^,Ll : (L/K)^ ,

where the symbol rr:rr denotes topol ogica'l i somorphi sm.

40

2. Some groups don't admit multipliers

Denote by lPthe circle group - the group (under

mul ti pl i cati on ) of al I comp'l ex numbers of rnodul us one (wi th

the induced topology), byR the group of real numbers, by Z

the integers, by Q the rational numbers and by Z(n) the

cyclic group of order n. For a fixed prime p, let Z(p-) be

the subgroup of 'll consisting of elenlents whose order is a

power of p, ÂO the group of p-adic integers and nO the group

of p-adic numbers (see Append'ix) .

LEMMA 2J. Suppose the abeLían group G has the

properl;y that for aLL x e G, 1;here eæis1;s an h e G such tha't

h2 = x. rf w is a nnútipLien on G srch thn'b å(x,y) = õ(y,x)

aLL x,Y e G, then u is triuiaL.

PR00F. õ(x,y) = o(x,v)/u(y,x) = ,(y,x)/r(x,y) = 12(y,x)

implies õ(x,y)2 = l all X,Y e G. Given any x,V. G, choose

h e G such that h2 = x, then õ(x,y) = õ(h2o.y) = õz(h,x) = 1.

That iS o is symmetric, thus trivial (1"1 (iii)).

LEMMA ¿.2, (Kleppner [25, Lemma 7.5]). Let G be a{..r^

disey,ete group uith muLtipLíer u. -rf õ(x,y) = õ(y,x\aLL

X,j e G, then u ís símiLar to a muLtípLien Lífted from a

quotient G/H uhich is of erponent 2 (that is eaeh e1'ement of

G/H has ordev, at tnost 2.).

PROOF. The mul t'ipl ier õ i s synunetrjc, so by 1.1 (i i i )

41.

there exists a function Y : G +'ll such that

õ(x,y) = Y(x)v(v)v(xy)-1. simi'larly, there exists a function

r : G +'lf such that r(x,y)o(y,x) = r(x).(y).(xy)-1 for all

X,! e G. Thus ur(x,y)z = vt(x)vt(y)vt(xy)-l for al1 x,y e G.

This'imp'lies t¡ is similar to a mu'lt'ipl'iêì trr, such that

11 = l. In fact ur,(x,y) = (yr(xy)vr(x)-Iyr(y)-1)"r(^,y),

where the square root is chosen in any fashion. Let H = Grr.

Since õr(x2,y) = rr(x,v)2 = l all X,y e G, lve have x2-e H all

x < G, that is G/H has exponent 2. Since ürr[HrH is trivial

and urf = 1, there exists t, : H +'ll such that

rr(x,v) = rr(x)tr(v)rr(xv)-I and rf is a character of H. Let

r' be an exterision of rl to G (t18, 24.12)), and defjne t, bV

tr(x) =t'(x)t'"h..e the square root is chosen in any fashion

with the restriction 11(x) = rr(x) aìl x e H. Let

urr(x,y) = ,r (x,y)"2(xy).2(*)-1.r(v)-1, then ,7 = l,rr(x,Y) = rr(Y,x) ali (x,y) e l{ x G and rr(x,Y) = l ali

xry e H.

Let, n be an irreducible or-representation of G and {9o}

a set of coset representatives rnodulo H containing the

identity of G. Define tto as follows

"o(x) = n(x), atrd

tro(xgo) = tt(x)n(go), for X e H.

Clearly n0 is a multipìier representation whose associated

multipl ier ur, is simil ar to ur, (and consequently o). Further-

more, for x,y e H,

no(xy9*) = "(xy)"(go)

42.

= TT(x)n(y)"(go)

= no (x)no (vso) ,

thus urr(x,y) = l all (x,y) e H x G, and by the symmetry of r,r.,

,r(V,x) = 1, (x,y) e H x G. From this fol'lows that ur, is

constant on the H x H cosets in G x G. Indeed, if g,h e H,

then o, (gx,hy) = ,, (g,x)u,¡ (gx,hy) = ,, (9,xhy),, (x,hy) =

or(x,hy) = tor(x,vh)or(v,h) = ra(x,y)r¡(xy,h) = rg(x,y). Thus

there exists a multìpìiêr t¡ron G/H whose lifting to G is

simi I ar to t¡. Thi s proves the I emma

Part of ífr. tollow'ing result can also be obtained from

[6, Lemma 2].

THE0REM 2.3. Let G be a discnete gz'oup uhich ís eíther

eycLic ot, of the form 7(p-) or Q,. rf u is a rruLtipLiez' on G,

then u is triuíaL.

PR00F. Case 1. Suppose G is cyclic. If it is finite,

say G = L(n), then õ(.,1), being a character of Z(n), is of

the form õ(p,1) = expl2nikp/n]; it now follows from the

bilinearity of õ that

õ(p,q) = exp[2nikpq/n]

p,Q e 7(n), k e 7. If G is infinite, then G is isomorph'ic

with V., and by the same reasoning as above,

õ(p,q) = exp[2nipqo]

43

p,g € V-for some G e [0,1[. At any raie õ is symmetric. The

only quotient of a cyc'lic group which is of exponent 2 is

Z(2) or the trivial group. Since a multip]ier r^r' on V.(2) can

be assumed to be normalized, we have 1. = o'(-1,1) =

o'(1,-1) = t,'(1,1), that is o' is trivial . Hence by

Lemma 2.2, u is trivial .

Case 2. The dual of Z(p-) is the group of p-adic

integers nO (see Appendix) which is torsion free. Thus the

homomorphism õ : Z(p*) - aO must be trivial. Hence õ = 1 and

õ is trivial.

Case 3. Q^ is torsion free and divisible (see Appendix).

If x e Q^, then for each n e 7, x + x(x/n), X e Q js a

character of Q wh'ich we denote by xlln and is the only

character of Q satisfying (vI/n)n = x (use the fact that Q^

is torsion free). Thus if the map õ: Q * Q^ maps 1 to x,

then õ( .,!/n) = xI/n, that is

õ(s/m ,t/n) = -[*^] t

this is symmetric, Q is divìsible, so by Lemma 2.1, r¡ is

trivial.

LEMMA 2.4. Let us be a Boz,eL rruLtipLier on a LocaLLy

contpact abeLian group G. If H is a dense subgz'oup of G such

that the restrietion of u to H is triuíaL, them u is triuiaL.

PR00F. õ : G x G -+'ll is continuous (1.1 (i)). The

44.

restriction õ : H x H +'[' is the trivial ffiðP, thus õ = 1 and ur

is rrivial (1.1 (iii)).

COROLLARY 2.5. A BoreL muLtipLíer on 67LA one of the

foLLouing groups.' 1ll lR, aO and ar; is tniuiaL.

PR00F. 7(p-) is dense 'in'lf, Q is dense in fR and nO, and

Z is dense 'in lO (see Appendix). Now use Lemma 2.4.

we know from I.1.3,(jii) that the dual of a torsion free

discrete group (for example Q^) is connected. The following

result shows that these groups clon't admit non-triviaì

mul ti p1 ì ers .

PROPOSITION 2 6 . Let G be a cornpaet {rou7. If G is

conneeted. (or equiuaLentLy diuisibLe (I.1.3.(iii))l then a

BoreL rm,LLtipLier u on G is triuiaL.

PR00F. The homomorphistn õ : G * G^ must be trivial

because G is connected and G^ discrete.

3. Some useful resul ts

THEOREM 3.1. Let G be a LoeaLLy eontpact abeLian

subgrotp uhieh has a cornpaet open subgt'oup K" St'ppose u is a

non-d.eganerate rm,tLtipLier' then u is type I if ' anÅ. onLy if G

eontains a eompact open rnaæí.maL ísctopíc subgzoup. MoreoÐev':

anA cornpaet maæírnaL isotropic H satísfi¿s õ(H) = A[G^,H]

aná G/H is topoLogicaLLy isomorphic to H^.

45

PR09F. Suppose ur is type I. By Proposition 1.5.(ii)'

lfu is compact and open, so K n lQo is a compact open isotrop'ic

subgroup. Without loss of generality, we assume Ko : K. Let<4|+? 4<'l--

H be a(mäximal isotropic subgroup, then õ(H) (being the

continuous image of a compact set) is compact and c'losed;

õ(G) is dense in G^, thus õ(H¡ = õ(Hur) = AIG^,Hl n õ(G)

= (AlG^,Hl n õ(G))- = AIG^,Hl n õ(e )- = AIG^,H1 , the fourth

equaìity being valid because A¡G^,H] is open anC closed. It

follows that õ restricted to H is a continuous isomorphism

onto A[G^,H] so it must be open [18, 5.29]. Let A be open in

G and A9 = ff n g-lH, g e G. The set A9 is open in H, thus the

set õ(A) - u{õ(g)õ(ng) : g e G}, being the union of open sets

in G^, is open in G^. The proposition now follows f,rom 1.3.

Let G be a locally compact abelian group for each

positìve integer n, denote by Gn the subgroup {gn t g e G}

and by Gn the closed subgroup {g e G : gñ = e}.

THEOREM 3 2" Let u be a non-degenerai;e tgpe I rruLtipLíer

on the separabLe abeLíon group G, then ufr is type | (n a

fiæed ù'tteger') if anå onLy if Gn is cLosed, in G"

PR00F. First observe that the proof of 1.3 extends

readily to assert that a multiplier ur (which is not

necessarily non-degenerate) on a (separabìe) group G is type

I if and only if the range of õ : G * G^ is closed. Let or be

as in the hypothesjs of the Theorem. By 1.3, G and G^ are

topoìogically isomorphic. The range of õn is (õ(c))n = (G^)n,

thus by the above remark, rr is type I if ancl only if Gn is

46.

closed in G"

THE0REM 3.3. Let G be a separabLe abeLian LoeaLLy eorrcpaet

gToLlp anÅ u a rm,LLtipLier on G. Thqn G6 ís l;ype I if and onLy

íf ^n is type I for aLL n e V-

PR00F. Observe that 'll i s a type I normal subgroup of G.

Si nce 'lI^ i s di screte, Go-orbi ts are cl osed, so by I .B .4, al I

factor representat'ions of Go are obtained using Mackey's

construction (I.8.5, I.8,6, I.8.7). Indeed, because 'll is

central , a factor representation n o'f= G0 reduces on Il to m.X,

where m is a iardinal anci x the character t * tn (t e n') of 'll

for some n e Z (see [15, IV.7.20])" Now m.x', where

X,(t,x) = x(t), ((t,x) n e') is a multip'lier extension of m.x

to Go, and the multiplìer associated with X' is precisely tr-n.

Thus by I.8.7, each n which restricts on 'll to a multiple of x

is type I if and onìy if r-h is type I, and Go is t¡rpe I if

and only if urn is type I all n e 7-.

Suppose G, ur and H are as in Theorem 3.1 and G is a group

of exponent p for some prime p (tfrat is each element in G has

order at most p). Since G is a vector space over the field of

p e'lements and H is a subspace it admìts a comp'letnent, that is

a subgroup K (isomorph'ic to G/H s H^) such that G = H x K.

Since H is open, we actuaìly have a topological isonorphism

between G and H x H^.

Such decornpos'itìons of G wher^e we 'insist that H is

compact and open cannot a'lways be found even if we ajlow the

47.

additionaì hypothesis that rh js type I for all n e î. (or

equivaìently that Go is type I), as the following exampìe

shows.

EXAMPLE 3.4. (i ). Let H be a locally compact abelian

Then o defi ned bygroupandG=H*H^.

(x,r),(y,x) e G is a non-degenerate multiplier which according

to 1.3 is type I. Moreover, if G is divisible and torsion

free, then on is non-degenerate and type I for each n e Z

(Theorem 3.2) ,

(ii). Suppose we let H in (i) be the group of p-adic

numbers r¿-, then H^ is aìso the group of p-adic numbers (seep-

Appenc!ìx), that is G = Qp * np, in particular G is divisible

and torsion free, thus ,n i, non-degenerate and type I for

each n < Z-. By Theorem 3.1, G has a tnaximal isotropic

compact open subgroup H. [{e show that. no such H can be a

direct summand of G. For otherwise, G must be isomorphic

with H * H^; but the dual of H, H being compact and totally

disconnected, must be' torsion (I.1.3.(ii)), contradicting the

torsion freeness of G.

(iii). For each t . Z, let K., be the group

7(4) = aç: C4 = 1) and Kj the subgroup Z(2) = (C2) of Kr.

Define H to be the subgroup of nK., consisting of elements

(ui) such that a., . Ki for all but a finite number of

indjces i; and topologize H so that nK.i (with the compact

cartesian product topoìogy) becomes a compact open subgroup.

48

Let o be as in (i). ClearlY, G2 is not closed in G and by

Theorem 3,2, w2 is not type I. Furthermore, Theorem 3.3 shows

that although o is type I, Gt is not a type I group.

Given a localìy compact abelian group G and a non-

degenerate type I mult'ipìier, we can however hope for the

existence of a cìosed isotropic subgroup H of G such that G

decomposes as a product H x H^. Indeed the rest of thjs

chapter is devoted to showing that such a subgroup exists under

certain addit'ional hypothes'is on G, for example if G is

divisible and separable, or if the connected component of G is

open. )

THEOREM 3.5 (Mackey [28' 9.6] ) . Let H and K be tuo

LocaLLy cornpaet abeLiarl groups artri u a rru.LtipLien H x K. If

ûr' is the ftmction on (H t K) * (H * K) defíned bA

r,r'(hk,h'k') = *(h,h')r(k,k')r(k,h'),

(h,k)(h',k') . H x K, then u' ds amuLtipLien uhich is siniLar

to u.

PROOF. For (h,k),(h',k') e H * K, we have

(,J(hk,h'k')r(h,k) = r(h,kh'k' )r(k,h'k'), and

,(h, h' )r(hh' , kk' ) = ,(h, h'kk' ) r(h' , kk' ),

thus

/1C

ûr(hk,h'k') =üJ h'k' û) h' TJ hh' kk'k h

û) ûJ

k' UJ h' û) k h'k',h' u) k,k ü)

Ih kû)

û) k

where Y is the function on H x K defined by Y(hk) = o(h,k)

Now

ûr(h',k' )o(h',k)"(k,h'k' ) _t,r(k;h' )'(k,k' )r(h', kk' )

h

kû)(¡

OJ

kk üJ

û)

kh'üJ

k h

,k=l

Hence the result.

COROLLARY 3.6. Let H and K be Loe.aLLy eompact and

abeLian. rf u is a rnn-degenez'ate rm'fltipliez' on H x K such

that Ku = H, then u is simiLar to the rruLtipLier w' defined by

t,t' ((h,k)(h',k' )) = r(h,h' )co(k,k' ).

PR00F_. Sìnce K,¡ = H, õ(k,h) = l all (h,k) e H x K.

Hence the result follows by Theorem 3.5.

If the condjtions of Corolìary 3.6 are satisfied, then

we say that o splits relative to H and K.

THEOREM 3.7. Let ube a non-degenerate BoreL muL'bipLier

on a LoeaLLy comapct gz'oup G. Suppose G has a marimaL

isotz,opic subgroup H uhich is a topologieaL direct sutnnanã,

then u ís type I if and onLg íf it is sinr|Lar to a

rruLtipLier u, of the forrn

t

,,(ri,(x,r),{,(y,x)) = À(y),

50

(x,r),(y,x) e H x H^, for some topoLogicaL isomonphisrn

{, : H x H^ + G.

ìl

PR00F. Suppose such an isomorph'ism ü exists, Denote by

i ts dual 'i somorph'i sm, then

v

where uro is the multiplier ro((x,l),(y,X)) = r(y),

(x,r),(y,X) . H * H^. Clearly õo is a topological isomprphjsm,

thus so is õ. .3, 0t and o are tYPe I.

Converse'ly, suppose ur is type I, then G 'is topologically

isomorphic with H x H^ (see proof of 3.1), thus we assume

without loss of generaìity that G = H ,. H^. Define

p : G + G^ by [p (x,r)] (y,x) = x(x)r(y)-, then p is a

bicont'inuous isomorphism such that p (H) = A[G^,H] ancl

p(H^) =A[G^,H^]. Itfollows thatthemapp-l o õ: G+G is

a bicontinuous automorphism such that p-l . õ(H) = H and

hence p-1 . õ(H^) = H^; but õ(H^o) = A[G^,1-l^] = p(H^) = õ(H^),

thus ll^ur = H^. By [1, Theorem 1], there exists a bicontinuous

automorphism p, (respectívely rpr) of H (respectjvely H^) such

that p-I . õ(x,À) = (Vr(x),{,r(x)) for aìl (x,r) e G. Define

{l : G + G by ,1,(x,r) = (,i;I (x),r), then rp is a bicontinuous

autornorphi sm.

Since H (respectively H^) is maximaì 'isotropic, the

restriction of o to H (respectively H^) is trivial. Thus by

Theorem 3.5, we can (and do) assume that o is similar to a

(¡) v o û)0 I

51

mul ti pì i er tr defi ned byI

t,, ( (x,l), (y,x) ) = õ( (t,r), (y,1) ),

(x,r),(y,X) . G. Clearly

,r({,(x,r),,t,(y,x) ) = õ( (1,À)), (q,;I (v),1))

= [õ(,t,;I (v),,l/;t (1 ) )] (1,r)

= [p(y,1)] (1,¡,)

= r(y).

Thus when'ever G decomposes as a product ll * H^ with H

maximal isotropic, we know precisely the form which ur takes.

A multipìieF r¡ satisfying the hypothesis of the above

theorem i s cal I ed a cross mul ti pì ier.

EIUA_Lp_. Let H, K and L be LocaLLy eonrpact anå.

abeTiqn øtd u a non-degenenate type I rm'LLtipLíer on H x K x L

urch that the map i: z H x K x L + H^ * K^ * L^ satí'sfies

õ(H) = ¡'^, then

õ(f¡ = H^, õ(L) = L^ arú. Lu: = K x H.

PROI|. By 1,5, K^ = õ(tl) = õ((Hr¡)t¡) = A[G^,Hur] , so

Ho = A[G,A[G^,Hr]l = A[G,K^] = H x L (use I.1.1.(ii)), thus

õ(H * L) = õ(Hr) = A[G^,H] = K^ * L^, consequently õ(L) = ¡^

and õ(K) - H^" Fina]]y o(Lo) = A[G^,t-] - H^ x K^ = õ(H >< K),

that is Lo = H x K.

52.

Yo &0.-9

The folìowing theorem allows ur{{uronot other things) #ffi with finite products of groups.

THEOREM 3 9. Let G be a LocaLLy conrpact abeLian gv'oup

uíth a non-degenerate and tgpe I muLtipLier u, and suppose H

is a cLosed isotropic subgroup uhich ís a topoLogieaL dírect

swmnanã. of G. Then G is topologicaLly isomorphic tLth

H x H^ x Ho/H and. u is simiLar to a rruLtipLier u' of the foyrn

r' ((h,À,X) (h' ,l' ,X' ) ) = . ((h,r)(h' ,r' ) )o(*,*'),

uhexe r ís a eross muLtípLier on H x H^ and" o is a non-

degenerate anÅ. type I nntLtipLier on Ho/H.

PR00l. All isomorphÍsms stated in this proof are

topo'logical isomorphisms. There exists a closed subgroup K of

G such that G = H x K; now õ(Ht¡) = A[G^,H] = A[H^ x K^,|-l] = K^

is a topological direct summand of G^. The nrap õ is an

i somorphi sm, thus þlo i s al so a topol og'ical di rect summand,

thus there exists closed subgroups K, L of G such that

H x L Ì Ho and G: Ho x K. Collecting all this information

together, we have

G:HxKxL

Hi,¡ËHxL

L = Ho/H

KÈG/HoãH^ (use 1.6 (iii)).

This proves the first part of the theorenl about the structure

53.

of G. Now o(H) = A[G^,Hur] = A[H^ t K^ x L^,H t Li = K^, thus

by Lemma 3.8,

õ(r¡ = H^, õ(L) = L and Lo = K x H.

It now follows from Coroìlary 3.6 that o is similar to a

multiplier to where t is a multip'lier on K x H and o is a

multiplier on L. Since

lr**

is a bicontinu'ous isomorph'ism from H x K to H^ x K^ such that

i(H¡ = K^ and i(t<) = u, by Theorem 3.7, r is a cross

multip'lier on H * K È H * H^. F'inaì'ly,

all ffill = õlr

is a bicontinuous isomorphism fr'onl L to L", thus o is type I

and non-degenerate on L Ì Hr/H.

COROLLARY 3 10. Let u be a non-degenenate ru.LtipLí'er on

the fìníte abeLi,an group G, then G is isomozphie to H x H^

for some subgroup H of G arld. u is simiLav.' to a eross muLtipLier"

!R001-, Ciearly o is type I. Using the structure theory

of abelian groups (see I.1.4.(iji)) G must be a finite product

e = II!=1 H.' where each H., is a cycl ic group of prime power

order. For each i, (Hi)o = {g e G : õ(n,9) = 1 all h e Hi}.

By Theoren 2.3, ,lH.i*Hi is trivial , so (Ht), = Hl . Now

HxKHxT1Õt)

K

54,

Theorem 3.9 and induct'ion yields the requ'ired result.

C0ROLLARY 3.11. Let u be a non-degenerate nuLtipLier on

Rn, then n is euen (n = 2k), u is type I anå. is a cnoss

muLtipLie, onRk * tRk.

PR00F. õ is a continuous homomorphism. Clearly

õ(r/s.x) = r/sõ(x) all F,s e v, x e Rr, ând by continuity

õ(t.x) = t,õ(x) , t efR, X .lRr. Thus õ is a vector space

homomorphìsm; since õ is jnjective, the range õ(fRn) is

n-dimensional and so is all of tRn. By 1.2 and 1.3, u is

type I . The r'est fol I ows from 3 .9 .

Recall that every localìy compact group G can be

written as a productfRn x H, where H is a locally compact

abelian group containing a compact open subgroup. The integer

n is an 'invariant of G (see I .I,2) ,

PROP0SITI0N 3.12. Let u be a ipn -degenerate and type I

mtLtípLíeo on lRfr x H uhev,e H is a LocaLT'U conr¡tact abeLían

g?ole, then u is simiLar to 'cç uhev'e r is a cross rm,fttípLien

on [Rfr anÅ. o is a non-d.egenenate and. type I nwLtipLíer on H.

PR00F. By Corollary 2.5, R is an isotropic subgroup. It

is a topological direct summand, thus by Theorem 3.9 and

induction, o is of the desired form.

55.

4. The connec'led com Donent of G as a dìrect summa nd

l^le saw in Section 3 that for our purposes, we can

disregard direct factors offR in G, so we can and do assume

that G has a compact open subgroup H. For the whole of this

section, we suppose that G adm'its a non-degenerate type I

multjplier o, Let C be the connected component of G, then C

must be a closed subgroup of H and is thus compact. We point

out at once that C is divìsible (I.1.3.(iii)); divisible sub-

groups of discrete abel ian groups are always direct summands

,

(I.1.4.(i)); furthermore, if C is a direct product of circle

groups then C is also a topoìogical direct summand

([18, 25.31a] ). Converse'ly, Fuìp and Griffith [13, Coroì lary

3.21 have shown that a connected group C which is a

topo'logical direct sum in every locally compact abelian group

(with compact open subgroup) in which jt occurs as the

connected component of the identjty must necessarily be a

product of circle groups. In particular, there exists a

localìy compact group G v¡hose identjty component is not a

topolog'ical direct summand of G. It is not clear if the same

is true jf G is self-dual , Below vre prov'ide some conditions

on G which ensures L,hat the connected component of the

identity is a topolog'ical direct summand"

LEMMA 4.1 . Suppose ue haue abeLian groLLps H g K s G

hn topoLogy) such thnt H is díuisibLe anÅ. K/H is a direct

sutftnanÃ. of G/H, thsn K ís q, diTect sutnnand. of G.

56

PR00F. From the hypothesi s and I .1 .4. (i ) , there exist

subgroups F and M in G such that K = H x F and 6 = fl x M.

Furthernrore, M can be chosen to contain F. Since F is a

direct summand of M, there exists a subgroup L such that

M = L x F. It follows that G = H x F x L and K is a direct

summand of G.

Let G be a locally compact group with compact open

subgroup H, connected component C and non-degenerate type I

multip'lier ur. since Cur is open in/(r.5 (ii)), Cu¡ is a

direct sunrnand of G in the discrete sense if and only'if it,

i s so i n the topoì og'i cal sense .

LËMMA 4 2 , Let G" u and C be as aboue, Then C is a

topologicaL dínect swnnand. of G if anrl onLy if Cu is a

dit ect swnnøtd of G.

PR00L. Suppose G is topoìogically isomorphìc to C x K,

then G'^ = c^ ' K^ and (1.5 (i)), õ(cr) = A[G^,c] = K^ is a

topoì og'ical direct summand. The "onìy i f " part now fol I ows

from the fact that õ is a topoìogical isomorphism. The "if"part is similar.

PROPOSITI0N 4.3. Let G, u and C be as aboue. If any of

the foLLouíng properties ave satisfied,

ft)(ü)

ft,ii)

G/C is tov,síon free,

CulC is díuisíbLe,

Cu/C has botmded order',

57.

ftu)

(u)

Cu/C is cornpact oz, C is open,

là is totsion free,

Curr/C is of erponent p lp - a fiæed prine), or

Curr/C ís eonrpact oz, C is open,

then C is a topoLogicaL di:rect swmnand. of G.

PR00F. (i). See Fulp [14, Coro'l1ary 9].

(ii ). Because Cur/C is divisibl e, it must be a d'irect

summand of GrC (I.1.4.(i)). Now C is divisible, so by 4.1,

Co is a direct summand of G and so by 4.2, C is a topological

direct summand of G. (An alternative proof can be constructed

usjns (i).)(iii ¡. Since (G/C)/(Co/C) = G/Cur which 'is topolog'icallv

isomorphic to C^, js a torsion free group (I.1.3. (i ij ) ), by

I.1.4.(ii), Co/C is a direct summand of G/C, Now proceed as

. ,..1rn (1rJ.

(iv). Since Co/C is self dual (1.5 (ìii)), Cr,r/C compact

or discrete 'impì'ies Co/C is f inite and thus of bounded order.

Now use (jii).(v). Since C is divìsjble it is a direct summand, thus

G torsion free implies G/C is torsion free. Now use (i).

THEOREM 4.4. Let G be a LoeaL Ly eonpaet abeLian group

and. u a non-degenerate and. tgpe I rmútipLier', then G is of the

fonm lRn x K, uhexe K eontains a cornpact open subgroup H

(I.1.2). Let C dernte the aormeeted component of K and u,

the xestriction of u to K. tf eíther

(+)

øi,)

5B

then G has a tnanímaL ísobz'opic subgnoup uhich is a topoLogicaL

dLv,ect summand. ConseqtentLy the stv'uchune of G and u ís

cornpLeteLy determined by Theoretn 3.7.

PR00F. By Proposition 3.I2, we may assume that n = 0.

Since C is compact and connected, Propositiorr 2.6 shows that.

C is isotropìc, thus by 3.9 and 4.3, we may assume G = Curr/C.

(i). As in 4.3 (ii'i), G rnust be fjn'ite. The result now

follows from Corolìary 3.i0. (ij). By Theorem 3.1, G has a

maximal isotrop'ic compact open subgroup L. Since G is a

vector space over the fìeld of p-e'lements and L is a subspace,

it admits a complement, but L is open, thus it is a topcl'logical

direct summand.

5. Local direct ducts and di v'i si bl e rou S

Given a locally compact abelian group G with non-

degenerate and type I rnultipìier o, observe that the group

Cu/C, where C is the connected componettt of the identity, is

a total]y disconnected group ([18, i.3]). l^le can deal v¡ith

some total'ly disconnected groups by decompos'ing them as local

direct products of topological p-groups. A definition of

local products is as follows. Let Gr''i e I be a collectjon

of locally compact abel'ian groups each with a compact open

subgroup Hr. The local direct product of the Gt with

respect to the compact open subgroups H.,, denoted by

LPi.I(Gi,Hi) is the subgroup of the full (d'iscrete) direct

product ni.IGi defined bY

59

{(Sr) e IIG., t 9i . Hi for all but finitely many i e I}

and is topolog'ized so that the subgrouP tri.IHj (with compact

cartesian product topo'logy) becomes a compa.ct open subgroup

of LP.,.t(Gi,Hj). (see also [18, 6.16])"

Abel ian topolog'ica'l p-groups (or simply p-groups, P ô

fired prime) and local direct product decompositjons of

topo'log.icaì groups into p-groups are dealt with jn deta'i'l by

Braconnier tsl and to some extent by Vilenkin [50]. The

following is a brjef exposition of tlre facts we need later.

An element x of an abeljan topoìogical group js called

p-prìmary if

xpn

lim _1

n-þ

(Brar,onnier's alternative equivalent definition states that x

is p-prj¡¡ary 'if the homomorphjsm Z + G : n -'xn extends to a

conti nuous homomorph'ism ao * G ' ) An abel ì an topo'logi cal group

consisting entjrely of p-p¡imary elements is called a p-group.

PROPOSITION 5.1, (tsl ) . A disere'be abeLían gt'oup G is a

p-group if and onLy if eaeh eLement of G has ordeT a pouer of

p

PR00F. The result follows immediately from the

defi ni ti on.

60

LEMMA 5.2, (t5l). A eonrpact abeLian group G is a p-grloup

if anå onLy íf G^ is a, P-Woup.

PR00F. Suppose G 'is compact. Let x e G^. For each

x e G, by the continujty of x, xPr * f imp1i., xpnl*¡ =

x(xP ) * 1, hence x(G) is a p-group; it is also a compact

subgroup of'lf and hence fìnite. In particular xpn = 1 some

n e 2.. Conversely suppose G is a discrete p-group, let U be

an open set in G^. By the definition of the topoiogy on G^,

there exists a finite set, - {xr, ..., *k} g G and e > 0

such that

{xeG^: lx(x)-1<eallxe FlsU.

n

Let n.1

be the smallest integer such that x

], then xPn

n = max{n

that xPn

=lallxeF,thusX= 1 and let

e U. It fol I ows

p

i1

np

t+LaSn-àæ.

PROPOSITION 5.3 ( t5l ) . Let H be a cLosed subgz'oup of

the p-groLtp G, then H an"d G/H are aLso p-gro'upri. ConuerseLy'

if H ís an open p-group in G and. G/H is a, p-group" then G ís

a, p-gToup.

PR09I. The first assertion about closed subgroups 'is

obvious. Let p : G + G/H be the canonical homomorph'ism. a

is continuous, so for all X e G, we infer from the'

djscreteness of G/H and Proposition 5.1 that ^Pn.

H for some

rì, so (*Pn)o* = *o**n * 1 ôS Ír + ø.

61.

THEOREM 5 4 ( I5l ) . Suppose the LocaLLy eompact abeLùcn

groLtp G has a contpact open subgz,oup H" then G ís a p-group if

aná. onLy ¿f G^ is a p-gtoup.

PR00F. Since G^/AIG^,H] is topo'logical'ly'isomorph'ic to

H^ which is a p-group (Lemma 5.2) and A[G",H] is isomorphic

to the dual of G/H which is also a p-group, the result

follows from Proposit,ion 5.3.

LEMMA 5.5. Let G be a LoeaL Ly eompaet abeLian gvoLLp. If

x e G is p-pr,ìmaty anÅ q-pnimarg for Luo distinct prinLes p and.

g, thenx=!.

PB00.F_. Let x . G^, then by the continuity of x,

x(*)Pn = ¡1¡pn) * 1, thus x(x) is a p-adic rational, that is

x(x¡Pn = l some n. similarìy, x(^)9* = l some m, thus

x(x) = 1. This is true for all x e G^, so x = 1.

PROPOSITION 5.6, (Braconnier [5, page 48]). Let G be a

conrpact totaLLy discormected group. Dernte by P the set of

rpimes {2, 3, ...}, then there eæist eLosed subgz'oups lr1,

(p . P) of G sueh that H, is a p-gnoup and

p€

Euz,therrnore, an eLqnent x e G beLongs to H^ for a prime p ifP

and. onLy if x ís p-prinarg.

PHpG=I

PROOF. By I.1.3. (ii ) G^ is a tors'ion group, thus by

62.

{[18, A3], ø is a weak direct product of p-Eroups. Now apply

duality and Lemma 5.2. For the second part, let vO denote the

canonicaì projection map n * tp and let x be q-primary (for a

fixed prime q), then ro(x) is both p-primary and q-primary, so

by Lemma 5.5, e,(x) = l whenever p I g. It follows that

x e HO.

THEOREM 5.7 (Braconnier [5, page 49] and Vilenkjn [50,

page 86] ). Suppose G is a LocaLLy contpact abelian gz'oup uith

cornpact open subgz'oLe H st'æh that both G anã. G^ are totaLly

discorm,eeted. Denote by P the set of pr'ímes {2r 3, ...},

then thez,e eælst subgz'oups Gr, Hp of G such that G, ís a

P-gTouP: H ís a eornpact open subgroup of G, anÅp

G = LPo.p(Gp'Hp) l{= np.P Hp

ELements of

that xln *GO are uniqueLy eharacterized as those x e G such

1.

PR00l. For x e G\H, define Hx to be the grorip generated

by {x} u H. The quotient G/H i's torsìon because-it is

topo'logically 'isomorphic wìth A[G^,H] ^ - the dual of a

compact totally disconnected group which must be torsion

(I.1.3.(ii)); thus Hx, the union of a finite nunrber of compact

cosets, is compact. Suppcse we have the primary

decomposi ti ons

H=I ^Hp€r pand Hx = Iro.p Hf

63.

which we may obtain by appealing to Propos'ition 5.6' then by

the latter part of that proposit'ion the group Gp = rr,.G Hf ls

a p-group and GO n H = HO. Since GpH/H is discrete, bY

[18, 5.32] nO/tO i s d'i screte and HO 'is open i n GO. Suppose

plqandx eGO nGO, thenLemma 5.5 shows thatx= 1. Next'

suppose x e G\H. Since Hx/H = np.p Hfluo is a finite product

(recaìl the index of H in Hx is finite), x is a finite sum

i X h+p€P p

with xO e Go and h e l-1. This says n =_loo.o(Gp'Hp). Finally,

the proof thad x e GO'if and only if xP * 1 is identical to

the proof of the corresponding fact in Proposition 5.6.

Having set up the necessary machinery, bJe proceed via a

few preì im'inary results to the majn theorem.

PROPOSIIION 5.8. Suppose G is totaLLy dísconnected anÅ

admíts a non-degenez'ate and type I rm,fltipLiev' u. Choose a

conrpact open manimaL isotfopic subgroup of G (3.I) ard Let

G = LPO.p(nO,tO) be a pninaxg decornpcsítion as in fheorem 5.7.

Ihen each G

rruLtipLierp , P € P adnits a rnn-degenerate and. type I

ÛJp

such that

fo? aLL (S.i ), (hi ) . G uhere 'the p?odtrct aboue is aLuays finí|;e.

a( (si ), (hi ) ) = rp.e õo (so, ho)

PROOF. Fix a prime q. Observe that G^ = LPP.P

64.

(a;, o tG;,Hpl ) i s precise'ly the primary decomposition of G "

and the image o(Go) consists of those ñ(x), x e G such that

õ1x¡en -> 1 as Íì à ær thus o'(Go) = G;. An.appeal to

Corol ì ary 3 .6 yi el ds the des i red mul ti P1 ì er ur' . It remai ns

to verify the above product formula. Let (St),(ht) . G and

let Q be a fin'ite subset of P such that 9O e ll', hO e HO for

all p ìn the complement of Q. Then G = L x K where

L = np.Q GO and K = LPp.pfQ(Gp,HO), and we can argue as above

to obta'in non-degenerate and type I multjpliers o and t on L

and K respective'ly such that t

Ip

and

However ;( (gi ), (hi ) ) = 1 s'ince gO

õ = õ and the result follows.

HO a1ì p c P\Q, thus

LEMMA 5 9. Let be a rpn'degenerate rm,LLtipLíez' on

nf.

ü).Q 'po 61

h €p

t1 - then

(i)

(ií,)

n = 2k, G is topoLogicaLLy isomorphie Lo K x K^

u'here K is a cLosed subgt'oup of G of tiæ form n\p

ctnð. u is cv'oss rm'tLtipLíer on K * K^. (rn

pæticuLaz', u is type I. )

If H is a conrpact open maæímaL isot:r'opíe

subgz'otp of G (anã sueh a gnoup aLuays eæists bg

Propos'ition 3.1.), then H is of the forin L x L'

for some eompact subgnoups L and l-' of K anã K^

respeetíueLy.

65.

!B!9I_. (i). Since G is a finite dimensional vector space

over the p-adic field n,,, the proof of Corollary 3.11 app'lies

to (i ).(ii). Let p (respectively rf) be the (continuous)

projection map G -' K (respect'ive'ly G * K^). First we show

that ç(H) s H and V(H) s H. Since ç(H) n V(H) = {1}' the

direct sum L = p(H) * {,(H) is a compact totally disconnected

group, so'its dual must be torsion (I.1.3.(ii)). Now õ is

injective and õ(t-1 = A[G^,Lo] ìs topologically isor¡orphic to

G/Lo and L^ (1.5), thus õ restricted to K must be trivial and

L isotropic. In particul ar õ(k,ø (h)) = õ(ç (k),,l, (h) ) ,

õ(,p(f<),ç(h)) j f for all h,k e H, thus ç(h) e Ht¡ = H whenever

h e H" Similarly with rl. It is now easy to check that

H=ç(H) "ü(H).

LTMMA 5 10 . Suppose ue. haÞe a separabLe LoeaLLy cornpact

abeLian p-grory u-hich is diuisibLe, If w is a non-

degenerate anÅ. type I rmt-LbípLier on G, then G ís of the form

nl far some integer n.p"

PR00F. Since G is self-dual, it must be torsion free

I.1.3.(i), thus by Rajagopa'lan and Soundararajan [36, Lemma 12],

G is a local product

G = LP.,.l(CIp,op)i

For G to be d'ivisibìe however, the index set I must be finite,

thus G i s of the form si!.p

66.

THEOREM 5.11. Let G be a Local Ly cornpaet abeLian group

uíth eornpact open subgroup anã. rÌþn-degenerate and type I

nwLtipLier u. Let C denote the eornponent of the i.dentity in

G. Sutppose Cu/C is sepaz,abLe ard. díuisi,bLe" then G is of the

form H x H^ aná u is simíLa:r to a, cross rruLtípLier.

PR00F. According to 2.6, 3.9 and 4.3 (ii), we can assume

without loss of generality that G = Cur/C. Let H be a compact

open maximal isotropic subgroup of G. Use 5.7 to write

ll= II ^HP€l' P

Now by Proposition 5.8, for each p € P, there exists a non-

degenerate and type I multiPlier ur' Gp

on such that

'G = LPPTP (no'to ) '

OJ np.P 'p

Observe that the hypotheses of Lemma 5.10 apply to each GO,

thus by Lemma 5.9, we can write

G = K * K'. l-lp p p- pL'

p

and K' = LPp-p(Kå,1.å),

=l_ xp )

where KO, Kå (respective'lV LO and Li) are cjosed subgroups of

nO (respect'ively HO) and KO 'is a max'imal isotropic subgroup

i n GO (rel ati ve to the mul ti p'l 'ier ,O ) . Let

K = LPpep(Kp'Lp)

thenG=KxK'and

67.

Kr¡ = {(s.¡). G: r((ht),(sr)) = l all (ht) e K}

= {(9i). G,np.p'p(hp,9o) = 1 all (hi) E K},

but KO is maximalty isotrop'ic in GO

Now apply Theorem 3.7.

for all p . P, thus l('¡ = K.

COROLLARY 5 T2 . Let G be a sepatabLe LocaLLy cornpact

abeLim groL{p. If G is ditsisibLe and adrnits a non-degenerate

anå. type I rruLtipLier u, then there eæists a cLosed subgroup

H of G and. a topohogieaL isomorphism front G to H * H^, such

that the image, of u wtder thís isomorphism ís simiLav' to a

rm,útipLier of the forTn

,r((x,r)(y,x)) = r(y) ,

(x,r),(y,x) e H x H^.

PR00F. By I.1.2 and 3.I2, we can assume without loss of

general'ity that G has a compact open subgroup. Since G is

self dual by I.1.3.(j) G is torsion free, thus by 4.3 (v)'

2.6 and 3.9, it must be of the form C x C^ x Co/C. Hence

C,¡/C is separable and divisible and the result follows from

Theorem 5.11.

Finally, we remark that Theorem 3.9, Proposition 2.6

together with [1, Theorem 2] show that the problem of the

structure of ur-type I locally compact abelian groups has been

reduced to that of 'res'idual' groups (see [1, page 597]).

68.

CHAPTER IIIl

MULTIPLIER REPRESENTATIONS OF DISCRETE GROUPS

Throughout this chapter, we fix a cj'iscrete group G and a

normalized multiplier ûr on G. Recall that. L2(G) denotes the

Hilbert space of square summable complex valued functions on G

w'ith scalar product ( , ) , B(12(G)) the space of bounded

l'inear operators on L2(G) and U(12(G)) the subspace of unitary

operators in B(12(c)).

We denote by r (respectìvely p) tfre ríght (respect'ive1y

I ef t) t, (respect j vely ur- I ) - representat'ion of G gi ven by

p,À : G + U(12(G)), where

(r(x)t) (g) = ,(s,x)f(gx)

(p(x)t) (g) = ,(x'1,g)t(x-19),

f e L (G),X,9 e G. To make sense of this definition, we observe

that

'¡(x) r'¡(v)!(s)

= J::ïï;;ïl:îl-,,' r( g,xy),¡(x,y) f( gxy)

= ,(x,y)[r(xy)f] (g),

IThe results contained in this chapter have appeared in l2}l.

ÂCt

all f e L (G), 9 € G, hence r(x)r(y) = ,(x,y)r(xy), and sirnilarìy

,(x,y)p(x)p(y) = p(xy).

Let V(G,o) (respectively V'(G,r)) denote the Von Neumann

algebra generated by p (respectively r), that js the weak closure

in B(12(c)) of the complex l'inear span of {p 9) : g e G}

(respectìvely {r(g) : g e G}). (See Chapter I, Section 2 and

Section 7. )

The a'im of th'is chapter is to investigate how various

statements about the maximal central type I proiectìon 'in

V(G,6) are reflected in the structure of the group G and the

multip'l'ier ur. Th'is leads to a characterization of ur-type I

discrete groups. The corresponding probleln for ordinary

representat'ions, that is if we assume o to be triv'ia'l , has

bee¡i successf ul ly deaì t with in Thoma [49] , Kani ut'h i22l ,

Smi th [45] , Formanek t10l and Schl icht'ing t4l I .

Our methods resemble more closely those found in Smith;

these are of a more elementary and algebra'ic nature than Thoma's

and Kan'iuth's "E(G)-Methoden" .

t. A representation of elements 'in V(G,o)

To begin, we construct a way to represent elements of t

V(G,r,r) as sequences in l-2(e ). For each x e G, denote by r* the

characteristic function of ixlc G. The set {eX

: x € G] is an

70

orthonornral basis for L2(G). We have

p(x)ç I = ).(x) -'r..e x

For a e B(L (G)), let a*,y = ( a(rr),r*) and a a.Xrê-X

x,y e G. The numbers ax,X € G are calTed the coefficients of

a.

LEMMA 1.1. (Kleppner 124, Lemma 1l). Let G be a disa'ete

gvol&, o a nonnaLized rm,¿LtipLier on G, ), the z'ight regular

u-vepresentatíon of G anã p the Left reguLar u-r-tepresentation

of G. Suppose â e B(12(G)),

(i)

(¿i)

ar(x) = r(x) a aLL X e G if and onLy íf

u*,y = o(x,y-i)axy- y, aLL x,y q G.

ap(x) = p(x) a aLL x e G if and onLy if

ur,y = r(y-t,x)ay- t* aLL X,! e G.

PROOF. Suppose a¡.(x) = r(x)a all x e G, then

ur,y = (a(vr) ,la*)

= ( a(r(y-l )r.) ,v*l

= ( a(r.) ,r(y)r(x- i )vul

= r¡(x,y-t)u*y-r,

all x,y e G. Converse'ly ìf a e B(12(G)) satisf ies

7i.

a = o(x,y- I )a 1al'l X,V € G, thenxyX,Y

< r(y)a(ç.),px) - 1 a(ve),r(y-r)r(x-1)pe)

= o(x,y)a a IX'Y xy-

- , u(rv-r),v*)

= ( aÀ(y)(r.) ,r*>

for all x,y e G, hence aL(y)(e.) = r(y)a(pe) aìl y e G. Itfollovrs that ar(y)(v¡) = r(y,h-r)ar(yh-t)(ee) = r(y)r(¡-t)a(ru) =

r(y)a(^f,), and hence that ¡.(y)a = aÀ(y) a1l y e G. The second

statement follciws from a sinlilar calculation.

PROPOSITION 1.2. Let G,u,). and p be as in Lemma 1.1,

Ifae B(12(c)), then

(i) ff r(x)a = ar(x) aLL x e G, then

u = In.G ano(s) ' and

(i¿) tf p(x)a = ap(x) aLL x e G, then

¿ = Ig.G agr(g-l),

uhey,e the suntnation is to be interpreted in ihe sense of ueak

operatoz, conuergence in B(12(G) ) .

PROOF. Suppose ¡,(x)a = aÀ(x) al I x e G, then

72

a(ey) = Ir.G u*,yt,

I*.e ,(X ,Y-l ) axy- t

Ig.6 '(9v,v-t)untn,

çX

Ig.6 r(v-I,9-r)ano(gY) ( ee)

= Is.G ano (s) ( tv) '

Thus the operators a and lg.e urO(S) agree on an orthonormal

basis in L2(G). It follows that tkiey are equal. This proves

part (i). Part (ii) ìs proved s'imilar'ly.

PROPOSITION 1 3. (K'leppner [24, Theorem 1] ) . Let

G,û,,p ond. x be as zln Lemma 1.1. Let a è B(L2(G)), then

(ì.) r(x)a = al(x) aLL X e G if and onLy íf

a e V(G,ur)

(ii) p(x)a = ap(x) aLL x e G if arui onLy if

â e V'(G,o)

(iii) Íhe eowmttart't Y(G,o) ' of Y(G,ur) is equaL to

V'(G,o) , Hence V and V' haue a cotftnon eentre

V N V"

PROOF. (i) Since

73

p(x)l(v)f(g) = o(x-r,g)r(g)f(x-19)

= r(x-I,g)o(¡-lg,y)f(x-lgy)

= r(x-r,gy)r( g,y) f(x-1gy)

= ,(g,y)p(x)f(gv)

= r(y)p(x)f(g)

alì X,Y,g e G,f u Lz(G) , we see that p and À commute' It

follows that each I(x) (x e G) commutes wìth all finite

conrplex linear combinations of the operators tp(g) : g e G].

But these linear combinations form a dense subspace o'f V(9,o)i

furthermore, multiplication is weakly cont'inuous in v(G,o),

consequently any a in V(G,ur) commutes w'ith each r(x) (x u G) .

Conversely, 'if ¡.(x)a = ar(x) , then by 1.2(ì)' a be'longs

to the von Neumann algebra generated by the operators p(g)'

g e G.

(ii) The proof of th'is part is similar to the a'bove'

(iii) If a e V(G,o)', then in particular ap(g) = p(g)a

aìl g. G and thus by (ij), a e V'(G,t). The remaining assert'ion

folIows from the definjtion of the centre (see defin'ition the

folìowing I.2.1).

We arra.nge some of thjs informat'ion'into a singìe statement.

THE9REM 1.4, Let G be a díscrete {roup, û a normaLised

m.tLtipliey. on G and. X the r,íght reguLar u-repÍ'esetztal;Ì'on of G'

74.

(i)

(ii)

sttppose ô e B(12(G)), then a e V(G ,u) í¡ ard.

only íf ar(g) = r(9)a aLL g < G. This oecuns if

and onLy if ax,y = ,(X,V-l)axy- ts aLL X,y e G.

rf a e V(G,ur) , then u =Lue ano(s) in the sense

of ueak operator conüergence. Ihis deconrpositíon

is wtique" that ís if a = IgeG a'no(9), then

u'g a all g u Gg

PR00F. Except for the utriqueness, all these assertions

follow immediately from 1.1, 1.2 and 1.3. Suppose we have

¿ = Ig.G a'no(ô), then a(,r.) = Ig.G u'grg and a* = ( a(r.),r*)

In.

,(x,y)uy-r*y = urr(Y,Y-1xY)

a all x e G.X

Since we are interested in central projectìons in V(G,o),

we need a method of determining when an element a e V(G,r)

be]ongs to the centre of V(G,o) 'in terms of the coefficients

ug (g . G) of a.

PROPOSITION 1.5. (KIep pner [24, page 557]). Let

Grtr tr and. p be as in Lemma 1.1. -If a e V(G'u) , then a ís in f;he

centv,e CV(Grr¡) = {b e V(G,o) : cb = bc aLL c e V(G,r)} íf and

onLy if

G u'g(rg,r*)

aLL x,y e G.

t

75.

PR0O.F. Consider the set of equivalent statements i

a e CV: ap(x) = p(x)a all x e G;

Ig.G uso(g)p(x) = IguG ano(x)p(s) ' all x e G;

I*.G uyr-rp(y)ur(¡-1,XY-r¡ = I*.G ax-rye(y)t¡(y-rx,X-1)'

all y e G; ar*-rr(V,x-1) = ô*-rn(x-I,v) all

r(x,y)uy-r*y = a*trr(y,y-Ixy) all X,! e G.

was obtained by a change o'F variable.

xrY e G;

The last equìvalence

g

PROPOSITI()N 1 (Kleppner [24, Theorem 2 and Lemma 6]).6

Ipt G be a discrete groutp and u a norrnaLized rm,¿L-bípLíer.

(i) Fon a e V(G,r), the mq I +â is ín L2(G).

(ii)

(io)

PROOF.

(ab)x = Ig.G uyby-r*r(y-I,x)

= Ir.G a*r-tbro(x,z-l) .

(äi) ( an,b/ = (ab*)", and (o*)g = ug-r.

Ihe map þ z V(G,o) * C : a + auis a finite

faithiuL nov,mal tv'aee on \,!(G,t)+ ' l;he set

of positi.ue elements in V(G,t,r) .

(ii), we have

(i) follows from (ììì) by letting a = b. For

76

(ab)x = (ab)*,. = I..G u*,=br,"

lrr; axr-rbzr(xrz-l).

The other formula follows by a change of va¡iable. The fìrst

part of (.iij) follows from (ij) and the second part from the

equation (ao)g = (a*(r.),r/ = (9.,a9/ = (avn,t.) = u.,g =

an-r:ur(e,g-1) = ug-, . The proof of (iv)is as follows:

Clearly O js finite and normal. If a > 0, then a = bb*

for some b e V(G,r); nor,r o = O(a) = â. = (bb"). = (llbnll¡z-(C))'

if and only if,a = 0. Thus 6 is faithful. For the invariance

we have O(ab) = (ab). =(ag,(b*)g) = (bg,(u*)g) = (bq)u = 0(ba).

Thus by I.3,2, V(G,o) js a finite votl Neunlann algebra (tfris

is not the case when G is non-discrete, see IV.4.1), thus

V(G,u:) has no type III, type II- attci type I- d'irect summands.

By I.3.3 V(G,r) is the d'irect sum of a type It von Neumann

algebra and a type II1 Von Neumann algebra. Indeed by I.4.1

there exi st central orthogonal proiect'i ons e,ê 1 rê2, . . . i n

V(G,o) wi th (li=, en) + e = I (ttre identity operator) and

such that enV(G,ur) ìs type In and eV(g,o) 'is type II1.

Let G be a djscrete groupr ür â normalìzed multìp1ier on

G and H a subgroup of G. For each t e V(G,r), let supp(a)

denote the support of a defined by supp(a) = {g € G: an I 0}.

Denote by V(H,o) the von Neumann aìgebra V(H,olHrH)'where

'lHrH is the restriction of o to H.

THEOREM 1.7. If ll is a subgz,oup of G, the set

77.

( = ia. V(G,r) : supp(a) c H\ is aueak operatov'closed

*-subaLgebz,a of v(G,r¡) (artd hence ís a uon Newnann aLgebr'ù, dnã'

there eæists a (normaL) *-isomorphism ' : V(H'o) * K'

Moleot)er', the coefficients of a e V(H,r.¡r) are presev'oed uttdert '

PR00F. Usjng the representation a = Ig.H ato(S) of

elements in K, we easily see that K'is a *-subalgebra of

V(G,o). To see that K is weakly cìosed' let {ai : i e I} c K

be a net converging weakly to an element a e V(G,t,l)' In

iparticular (ar(ç.),erl *(a(e.),er); but for x { H,

< ui (r.),rx) = 0 all i, hence u* = ( a(.r.),r*) 'is also equa'l t'o

0. ,

Let S be a set of coset reptesentatives modulo H'

A *-'isomorphism ' : V(H,6) * K can be defineci as follows

a'f(x) =

f e Lz(G); where g € Sn h e H are two elements such that X = hg'

and fn denotes the function h ' f(hg)r(h,g), h e H'

Note that

o(g,h)fgx(h) = f(hgx),¡(h'gx),¡(g,x)

= f(ghx)u,(hg,x),¡(h'g)

= f*(rrg)u,(h,g)

= r(g) fx(h) ,

that is r,r(g,h)f gx = f(g)fx. Hence ìf hg=h'g', h,h'e H'

7B

9,g' e G, then

(À(h- rn' ) (afû) r9 )ur

, )l (h)

This shows that the definjtion of ' does not depend on the

choice of the transversal S. Moreover, we have

lla'fll L2(c) = IgeS,h.Hl (arn) (h) l2

Isus(llafnll ¡z 1¡¡)2

ll all z [nur,h.Hl fg(h) l2

L2(G) )2

ô e V(H,r), f e L2(G). Hence a'e B(12(G)) for a'll â e V(H,r).

The next step'is to show that the map ' indeed maps into K. We

do this by caìculating coefficients. Let h e H,9 € S and

X,y€Gbesuchthatx=hg,

= il all2 (ll f ll

79.

t(çv g

Hence,

Suppose a' =

so jn partjcular,

Yg-tr¡(g,h-1),0

y-lg e l{,otherl',¡i se.

) l(h){

a'xy = < a'(çn) 'çl = a'(çr) (x)

In particular u'* = ô,at

v g

0 otherwise

xy-rür(x,y-I) if xy-t e H,

otherw'i se.

x e H. Also a'

I (h)/r(h 'g)

V(H,al) , then a'*

H which implies

a[(ç

la(rnn 1)l (rr),(g'v-r)/"1rr,g¡ if vg-i ' H'

0 otherwise

uhgy- rr(h, gy- I ),¡( g,Y- I )/r( h, g ) ifv Hg HX,€

a -rt,r(x,y-l),Xrê X'J xy

0 otherwi se

hence a' e K for all a e V(Ìl,ur). S'ince multipì'ication and

involution in V(G,o) and K are defined wholly in terms of the coefficients

a.., respectively ô'x (which are preserved undeF'), we see that I

X

is indeed a *-homorphism from V(H,r) into K.

0 for SOme a €

all X e

0

a=0allXeG,

, thus theâ*=o

80

map' is injective. To show that' is onto, we construct a right

inverse " for', that is amap ": K+V(H,to) such thatro " - idK.

If f e L2(H), we denote by f the function

f(g) e e H,f (g) =

0 otherwi se.

Defjne " by a"f = a(f )lH, f e L2(H), a e K. Since

lla¡rfllL2(H) = llaf l¡ll¡z(ti).

< llafl llL2(G)

= ll all ll fllL2_ (t-t )

as well as a"*,y = < a[(çr)" l,r*) = âx,y = o(x,y-t)u*y-r =

r¡(x,y-l) â"xy-1 for X,y € H, we see that " maps into V(H'r).

That I o " = id* foì]ow by comparing coefficients. The

asserted nornral 'ity foì I ows f rom | .2.2.

In view of this Theorem and because we are irrterested

only in the type structure of V(G,r¡), we rvill identify V(H,o)

with the weakly operator closed *-subalgebra {a e V(G,r) : supp(a) c H}

whenever H is a subgroup of G.

2. Di screte f i n'ite cl ass qroups

Recall from Chapter I Section 7 that a (discrete) group G

81.

is called a FC group (finìte class) if the set {g-1x9 : g e G}

is finjte for each x e G; and for an arbitrary (d'iscrete group)

G, GFC denotesthe subgroup of ail elements x in G such that

{g-lxg: g € G}'is a finite set. Clearly GfC ís a FC group.

For subgroups H and K, let [H,K] be the subgroup of G generated

by {n-tk-thk : h e l-1, k e K}. The group [H,H] 'is denoted by

H'. Itis a non-trivial theorem of Neumann ( ¡aZ, Theorem 5.11)

that for finite class groups G, the commutator G' is locally

finite (tnat is, each fin'itely generated subgroup of G' is finite).

hle need this result in the form of the following'lemma.

LEMMA 2.1 (Neumann [33, Lemma 4.1]). Let G be an FC g?ottp,-----.-.--------

H a subgroup of finite índeæ in G such that lH'I a -, then

lc'l < æ.

PB00r_. Let S be a set of coset representatives modulo ].l.

If s,t e S and g,h e H, then by using the equalities [a'bc]

¡a,cl ¡a,blc, [ab,c] = [â,c1b¡b,c1 (where [a,b] denotes a-lb-1ab

and ab denotes U-Iab), we obtain

[gs,ht] = [9,t]slg,hlttIs,t] ¡s,hlt ,

so we see that G' ìs generated by elements of the forrn [s'h],[s,t],

[g,h] and the'ir coniugates. The commutator H'is finÌte by

hypothesis, so there are only finitely many eìenlents of the form

[g,h]. Since S js a finite set, there are on'ly finitely many

elements of the form [s,t]. To see that there are on'ly fin'itely

many eì ernents of the f orm [s , h] , note that [s ,h.l = s- lsh , but

G is a FC group, so we have the desired property. h'e have shown

8?.

that G' is finitely generated. The result now follows because

G' is locally finite.

3. The o-finite class qroup and the ar-centre of G

Let G be a dìscrete group and t¡ a normalized multiplìer on

G. The concepts of finite class group, centre and centralizer

of G have thejr natural analogues for the pair (G,o). We

define these and invest'igate some of their properties.

Let l-t be a subgroup of G. For each x e H, the ur-centralizer

of x in H 'is defjned by

Cr,¡(x) = {g e cr(x) : ur(x,g) = o(g,x)},

and the o-centre of H by

Z,(H) = ^*.H Cr,r(x)

= {g € Z(H) : o(x,g) = o(g,x) al I x. e H},

where Cr(x) denotes the (ordinary) centralizer of x in H and Z(H)

the centre of H.

The o-finjte class group of G is defined by

A= an = {x e G : [G : Cu,,6(x)l < æ]

Note that if o is a trivial multìplier, then  = GFc. Because

Cr,nh)c- C(x) each x, we have ¡ 9 GrC. In particular ¡ is a FC

group. The fol I owi ng propos'i ti on wi I I i ust'i fy the abrove def j ni t'ion .

83

PROPOSITION 3.1. Let G be a discrete gt'oups 6 4 norTnalized

ru,LtipLiev'on G and p the Left TeguLaz' ,-r-t'ept'esentation of G.

LetxeG,then

Cr(x) = Cr,dx) = {g e G : p(x)p(g) = p(g)p(x)1.(1'.)

(ií) Cr(x) and A are subgroups of G.

PR00F. (i). Suppose g . Cr(x), that is g e C(x) and

o(x,g) = ,(g,x), then p(g)p(x).(g,x) = p(gx) = p(xg) = p(x)p(g)

o(x,g), so p(x)p(g) = p(g)p(x). Conversely, suppose g e G

satisfies p(xÍp(g) = p(g)p(x), then p(gx)/'(g,x) = p(xg)/,(x,s),

so in particular,

lp(gx)r.J (v) Ip (xg )v*l (v)

û) 9'X r¡(X,g )

for all y. G, where.r" denotes the characteristic function

of {e}. Thus

to(x-tn-1,V),/"(x-tg-tr1 o(g-1x-1,y)r.(g-Ix-Iy)T b-G'gl--

Now if we let y = gx, this expression simplifies to

,(g,x)-l = o(9-1x-1,9X)r.(S-t*-ty)/o(x,g), hence vu(S-tx-ly)

cannot be zero, but the only way th'is call occur is if gx = xg

This shows that g e C(x) and from the above expression,

r(x,g) = r(g,x). This completes the proof of part (i).

(ii). Suppose h,9 € cr(x), then

B4

p(h-1g)p(x) = p(h)-rp(g)p(x)r(rr-t,g¡-t

= p(h)-rp(x)p(g)r(h-t,91-t

= p(x)p(h)-Ip(g)r(h-r,g)-1

= p(x)p(h_ig),

hence h-rg e cr(x) and Cr(x) is a subgroup of G

cr(x-ly) > cr(x) n cr(v), we have

Si nce

tG : cr(x-ry)l < tc : cr(x) n cr(y)l

(-,

whenever X,y € A, hence ¡ is a subgroup of G and thìs proves

(i i ).

The fol'lowìng theorem po'ints out the significance of ¡.

THEOREM 3.2. Let G be a &Lscrete gï,oup, w a. norrnq.Lized

rruLtípLiet,on G anå. n the u-finite eLass gvoup of G. ff V(G,ur)

'h.a.s a non-zero manímaL type I Pæt' thert

(i) [G : A] ( -, and

(ii) l¡'l < @.

The idea of the proof of (i) comes frorn Smjth [44, Theorem 9.4],

and the proof of (ii) 'is s'imilar to that of Snljth [45, Theorem 1].

hle need the folìowing lemmas.

LEMMA 3.3. Let V be a tApe In uon Neumann aLge.bra, then

fon each non-zero írreducibLe z'epresentation n of V, we have

n(\t¡ = Mn(CI) (the space of n x n matvLces oler, E). Hence the

dimension of r equaLs n.

PR00F. Let Srn denote the standard po'lynomial of 2n

variables (see Chapter I Section 4). By i.4.5, S2n(V) = tOÌ

and Srn("(v)) = n(S2n(V) = {0}, Let Hn denote the Hilbert

space of r. It follows from I.5.1 and the jrreducjbility of

n that the weak operator closure of {n(a) " a € V} in B(Hn)

equaìs B(Hn), thus by I.4.6, Srn(B(Hn)) = 0. We conclude

(us'ing i.4.3) tfiat the cljmension of n is not greater than n.

V is an n x n matrix algebr'a over the centre CV of V

(see Chapter I Section 4), that is each a e V can be written

a = Ir<i,j=n oijcij '

where o.ij u CV and c' denotes the n x n matrix whose onìy

non-zero entry is the i,jth entry which consists o'f the

identity operator in CV. Fix j,k e {1, , r}. Sínce

.ij.jkcks = crn, all L < i,i < n, n(c¡¡) = 0 ìmp'lies n(cr.s,) = 0

all 1 < i,j < fl and hence "(V) = {0} which is a contradiction.

Thus n(cr¡) I o att i,k. Now suppose we have complex numbers

ßij, 1 < i,i < h such that Ii¡ß¡jn(cij) = 0, then for any

fixed k,,c. (]. < k,.[ < o),

Ii , jßi j"(CLt Ci.jCuu) = ßkl,n(Cou) = 0,

B6

this implies ßOu = 0; in other words, the tt(c¡u), 1 < k,t < h

form a basís for n(V), thus n(V) = Mn(CI) = B(Hn).

LEMMA 3.4. Let G be a discr.ete groLtp and u anorrnaLízed

rm,útipLier on G. Supposa CV(G,u) denotes the centz,e of V(G,ur) ,

then CV (G,r) c V(^,ûr).

PROOF. Let a e CV(G,o) and X e G such that a* I 0,

then by I,7, it suffices to show that X e a. From 1.5,

u¡(x,y) uy-r"y = a^.(v,y-1xy), thus C(x) = Cr(x), but by 1.6('i),

[G: C(x)] . -, hence X e a.

LEI4MA 3.5. Suppose G is a discrete FC groups a e

namnalízerl rruLtipLier on G, p the Left reguLar u-repv,esentation

of G and H a subgz,óup of G. Suppose there eæists a pz,oiection

e in CV such that both eV and e\f are type In (n a fiæed ínteger) "

uhere V arLd \f denpte 1;Lte uon Neumøn aLçTebras V(G,r,:) and. V(H,ur)

z,espeetiueLy. rf K= {xe G: p(h)p(x) = p(x)p(h)aLL he H}=

^¡,.H Cf,,De(h) ¿s the u-centz,alizer of H ín G, then e(p(k)p(g) - p(g)p(k)j = o

for euery k e K and g e G, Consequent_Ly [K,G] is fi.nite.

PR00t. Let n be a non-zero irreducibìe representation of eV.

Since n(ea) is non-zero for some a e V, n(e)"(ea) = n(ea) I 0, so

n(el) = r(e) I O, but eI e eV", thus r restricted to eV' is non-zero.

By Lemma 3.3, n(eV" ) = Mn(C) = n(eV).

If k e K, then ep (k) central'izes eW , so Tr(ep (k)) central izes

n(ef ) = n(eV), 'in particular n(ep(k)) commutes with n(ep(S)) for

87.

every g e G, that is "[e(p(k)p(g) - p(s)p(f ))l = 0. By I.5.3,

ep(k)p(g) = ep(g)p(k). If we write e = Ig.Geno(o) (1.4), then

f*.G.*o(x)p(g)p(k) = Ix.Ge"o(x)o(k)p(g)

that is

Iyuceyk-rn- r,(s, k) - Ir¡(gk,y- I )p (y) =

IyuGevg-Ik tr(k,g)r(kg,Y- I )-lP (Y).

Equating the coefficients (using the uniqueness of decompositÍon)

and letting x = kg gíves lutnL-ro-rl = leri. By Proposition 1.6(i),

iluglz < -, so the set ikgk-tg-t ' k e K,g € G] must be finite.

Since this set generates [K,G] and is contained jn the ìoca'l]y

finite group G' , 'it fol lows that IK,G] is finite.

PR00F 0F TllE0REl,1 3.2. (i) Let en be a non-zero central

projection in V(G,ur) such that enV(G,o) 'is type In. Since

enV(G,ur) is a matrix algebra over its centre (I.4), it is of

dirnension at nrost n2 over CenV(G,r) = enCV(G,r) g CV(G,t',) ç V(¡)

(Lemma 3.4). Hence if gl, ... , gn2+1 are n2 + 1 elements of G

beì ongi ng to di st'inct cosets of ¡, then there ex'ist el ements

cr, , cn2+t e V(n,tr) such that

n2+ t n2+ tIi=r .i(e,.,0(9i)) = i1=1 cieno(9i) = 0.

with not all (cien)o(Si) = 0, but ciên. V(l,ur), so this cannot

88.

happen since the sum

V(¿,ur)p(gr) * ... o V(a,o)o (9nznr).

is a direct sum. This shows [G : A]'n2 < -.

(ii) Again we use the standard poìynomial in 2n

variables Srn (I.4). Suppose V(G,ur) has a non-zero type I

part, then there exists a central proiection en I 0 such

that enV(G,r) Ís type In. l.le have Srn(.nV(G,r)) = {0},

hence Srn(enV(¡,r)) = {0}. It follows that enV(a,r,r) is

type I.n (I.4.5)

Now let en be a non-zero central proiection in V(¡,o) such

that enV(¡,r) is type In. Using I.4.5'

Srn(.nv(¡,r)) = o, Srn_ (env(l,o)) f o.

Since the polynomial is multilinear and V(¡,r) is generated

by the elements p(g), g e a, there exist 91' , 92n-2- e a

with srn-r(.no(gr), , €np(9rn-r)) * o. Let H be the

normal subgroup of I generated by the eìements 91, , 9zn-2

and the'ir coniugates. Since A is a fin'ite class group, H is

finitely generated. Moreover, Srn(enV(H,r)) = 0 and Srn- (enV(H,,) I 0

so by I.4.5, .nV(H,r) has a nonzero type In direct summand (.,t.o<ø' '*^ù,UeV(H,o) for some e e CenV(H,r). We wish to show that e e CV(¡,r).

Since H is normal in ^,

for any k e a, the automorphism

V(H,o) + V(H,ur) : a + p(k-i)ap(k) leaves the type In summand

ev(H,ur) fixed, that is p(k)-1ep(k)v(tt,r)o 5o by the unìqueness of e'

p(k-l)ep(k) = e. It follows that e e CV(A,o). Sjnce e r ênr

B9

eV(A,o) is clearly also type In, and thus by Lemma 3.5, K'< @'

whereK='{xeA: p(h)p(x) =p(x)p(h) all he H}. SÍnceH js

finitely generated [A : K] < - and thus the result l¡'l '-follows from Lemma 2.1.

Theorem 3.2 provides us with a subgroup of G of finite

index'in G whose comrnutator subgroup is finite provided V(G'r)

has a non-zero type I part. In the next chapter' We construct

such a subgroup for arbitrary local]y compact groups, G using

the resul ts of Tayl or t48l . However, si nce Tayl or's work

depends on the results known for rliscrete groups (and s'ince our

proofs are of an elementary nature), we feel iustified in

i ncl ud'ing them here.

4. The type I part of V(G,t¡)

Let G be a discrete group and o a norrnalized multjplier on

G. Recall that the group extension Go of T by G is the group

whose under'lying set is T x G a.nd with multip'lication

(s,x)(t,y) = (sto(x,y),Xy),(s,x)(t,y) e G'. usually Go is

endowed wi th the wei I topo'logy ( i .6 ) . Holvever, for the re-

mainder of this chapter, we give Go the d'iscrete topology.

l^lhenever H is a subgroup of G, we identify H'in the obvious way

with a subgroup of G0.

PROPOSITION 4.1. Let H be a subgrottp of the discrete

group G and. $ q, TlormcrLízed rmfltip'Lier on G, Adopt the vLotation

of Section 3, then

(i) c6u,(t'x) = (cr(x))'fov'aLL (t,x) € G',

90

(í'¿) [G' : H'] = [G : H] ,

(¿¿i.) (G')rC = 40, and

(iu) (2u,(G))' = z(G').

PR00F. (j ) Suppose (s,y) e C6o(t,x), then (stur(x,y),xy) =

(sto(y,x),yx), thus y . Cr(x) and (t,y) . (Cr(x) )'. Conversely if(t,y) . (cr(x)), then y e cr(x), so (s,x)(t,y) = (t,y)(s,x).

(ii ¡ The map xl-l * (t,x)H' sets up a one-to-one

correspondence between the H cosets in G and the Ho cosets

in G0.

(jii) By definition, each X e G belongs to ¡ if and

only if [G : Cr(x)] . -. Bv (i ) th'is occurs if and only if

[Gt : CGo(t,x)] < * âll (t,x)u nu. But thjs last statement is

equivalent to (t,x) . (G')fc.

(iv) Fix x e G, then (t,x) € (zû,(c))'u if and only

if x e Z(G) and ,(x,y) = ,(y,x) all y e G, that. is if and only

(t,x)(s,y) = (s,y)(t,x) all (s,y) e G0.

Recall that for any discrete group G, Go denotes the

von Neumann kernel of G. (For a. definition see Chapter I

Secti on 7. )

LEMMA 4.2. Let G be a disenete group and u a norrnaLized

rruLtípLíer on G. Suppose that H is a subgnoup of firúte

91.

indeæ in G such 1;Vnt lH' | < - then there eæists a subgnoup K

of G such tVnt tG : K:l < - artÅ. K' = Qo = {L' : [G : L] < æ].

PR00F-. If tG: Ll < - then, since the characters of'

L/L' separate points ([18, 22.17]), we have G" = Lo I L', hence

& cñ {L' : tG : Ll < æ}. Since lH'l < -, ¿nd H/l-L is

maximally a'lmost periodic, by I.7.3 and I.7.10, H/H" is type i

and H has a subgroup. K containing H" of finite index'in H such

that K/H. is abelian. Since tG : Kl < - ênd K' c Ho = è,ihe result follows.

The next iemma is a key lemma.

LEMMA 4.3. If thez,e eæísts a subgroup H of G sucVt that

tG : H] < @ lH'l < - and. tnlHxH is tt'iuíaL fo' oorne n, then

thez,e eæists a subgroup K of G sueh tVøt tG : K] ( -, 6 = K'

and (Gu). = (K')'.

PROOF "

that H' = G.

all X¡y e H.

By Lemma 4 .2, we can as s ume wi thout I oss of general ì ty

For some map y : H + T, u,h(x,y) = y(x)v(y)/v(xv)

An easy calculation shows that

(H')' = {[o(x,y)r(x-I,y-t)r(xy,x-ly-l),xyx-Iy-i] : X,y e H].

Si nce

lt¡(x,y)t¡(x-I,.v-1 )r(xy,x-Iy- t )ln = y(xyx-Iy-t )-1,

we have (H')' < æ. By Lemma 4.2, Go has a subgroup Ntf.rt ,rlua

[G' : M] < @ and M' = (G').. Let L be the image of the proiection

9?.

M + G : (t,x) -n x. L is a subgroup of G with the property

M c Lo, hence [G' : L'] < @; furthermore, (L')' = M' and thus

K = L n H has the desired properties.

The fol I ow'i ng theorem characteri zes expì i ci tly the

maximal type I central projection 'in V(G,o). It is the main

result of this chapter.

THEOREM 4.4. Let G be a discrete group uith normaLized

rruLtipLieruandLete (r,espectiueLg en, tt = 1, , n l-)be the masímaL type L (respectiueLy type In) centnal proiectior"s

in V(G,u), t?rcn

I n<æ ne -e

and the foLlouíng az,e equiuaLent.

(d el0(b) thene eæis'bs a subgz.oup H of G such that

tG : Hl < * and lH'l < * and rlHrH ís tz'iuiaL,

(c) IG : Al . -, lA'l ( æ and G adnits a finitedimensionaL u-z,epresentation. (l d.eno'Les the.

u-fini'te eLass group of G.)

Suppose, e I 0, t\ten there eæists a l-Cimensioy¿aL

ur-vepresentation y of G, sueh that

nlç = (d'im n). ^,(, for aLL finite &imensionrtT.

.,o-represenbations n of G.

(i)

93.

(¿.1) I v(g)p(g).e= ge G.

PR00F-. Suppose e I 0, then en I 0 for some integer n.

Lemma 3.3 ensuresthat an 'irreducible representation t of

enV(G,ur) will give rise to a finíte dimensional o-representation

r : g + r(eno(g-l))* of G, where A* denotes the adjoint of A.

Together w j th Theorem 3. 2 , thi s yi e'l ds (c ) .

Suppose we have (c). Let r be a fin'ite dimensional

o-representation of G. After taking determjnants,

,n(x,y)det n(xy) = det n(x) det n(y), and we see than ,fl is

trivial, to úy Theoren 3.2 and Lemma 4.3, there exjsts a group

Ksuchthat tG: Kl <-Ço)K', lG.l <-ând(G')"= (K')'. Let

n be a.n irreducible finjte dimens'ional o-representat'ion of K,

then no: (l,x) + rn(x) is a finite dimensional representation of

Kûr (I.6), so no (1,x)(1,y)(1,x)-1(t,y)-1) = 'r(x)n(y)u(x)-1r(y)-i = I

all X,y € K, but n is irreducible so it is one dimens'ional (see

remark following I.5.1), consequently trlfrf is triv'ial. This

implies (b).

Finally suppose (b) 'is true. Since urlHrH is trivjal, H has

a one-dimens jonal o-representat.ion, hence by 'induc'ing (I.8), we

see that G admits a finite dimensional ur-representat'ion. Let n

be such a representation. Lemma 4.3 is applicable, so as'in the

preceding paragraph, n(x)n(y) = tr(y)n(x) al I X,Y € K, and â = K' ,

where K is the subgroup we obtain from Lemma 4.3, consequently

In(xyx-ly-1) = r(y,x)r(y-1,¡-1)r(yx,y-1x-1).

94.

Since the lefthand side of this expression is independent of

the way we express xyx-ly-l as a commutator, and since the

righthand s'ide depends only on the dimens'ion of n, the functìon

y : xlx-In-t + o(.y,x)r(y-1,¡-1)tr(yxy-lx-l), X,Y € G, extends

to a well defined o-representation of G that satisfies (i).

To complete the proof of thjs theorem, we must show that our

cument assumptions lead to the statement (a) and (ii ). Let

-LFr = lËT Inuav(g)P(g)'

By Propos'iti0n 1.5, f is central in V(G,o). Since g + Y(g)p(g)

is an ord'inary representation of K, f2 = f. We cla'im that fV(G,tu)

is abelian. Suppose a,b e V(G,') and n is a fin'ite d'imensional

o-representatjon of G, then

(fba), = I*ï lru*,v(z) (5a)z-.rro(z-r,x).

= Tl*T Iru *, Ir. KY ( z ) a.ubr- t *y- r' (z- t x,y- I )' ( z- I, x )

= Él f r. *, Iy. Kaybr- t xy- ir ( x) n (v ) t ( z- r xv- r ) - r

Simi I arly,

(abf)* = ¡*\ Ir.*, Ir.ruyby-r*r-r'(x)n(y)n(y-1xz-1)-1

Since K' is normal in G and xy-rx-ly. K', {y-txz-I : z € K'}

= {z-rxy-r : z €. K,}. It follows that (fa)(fb) = abf = fba = (fb)(fa).

If we write fV(G,o) as a module direct sum

95.

fV(K,ur)p(91) @ ... o fV(f ,r)p(gk)

for some set of coset representatives 91, , 9¡ modulo K,

then by representing fV(e,r) as right mu'ltiplication on

itself, fV(G,o) is a matrjx algebra over the abelian algebra

fV(K,o) and thus by I.4.2 and I.4.5 is' type I. Thjs proves (a).

By iook'ing at the irreducible representations of enV(G,to),

whenever this is non-zero, and using I.5.3, we see that eV(G,r)

has enough finite dimensional representations to separate the

points of eV!n,r). However n is a non-zero fin'ite dimensional

representation of eV(G,r), g + "(p(g-t))* is a finite dimensjonal

ur-rêprêsêntation of G, hence by ('i), n(p(S)v(g)) = I all g e è = K',

thus

,,(f ) = Þi Is.G ,,(p (g)v(g)) = I,

from which we obtain n(e - f) = n(e) - n(f) = 0. Since n is

arb'itrary, we conclude that e = f. This completes the proof

of Theorem 4.4.

Note that for t¡ trivial, this theorem reduces to results

due to Formanek [10, Theorem 2] and Schlichting [41, Satz L].

The fol'lowing theorem is a consequence of 4,4.

THE0REM 4.5. Let G be a &Lscete grory and u a

norrnaLízed muLtipLíer on' G, then the foLLoaíng are equiuaLent,

U,) V(G,ur) is tgpe I (or equiuaLentLy type Iç),

96

(ì.i) euetaA u-representatíon of G is type I,

(Ì,ii) G has an abeLian subgz'oup A of' finíte indeæ

in G sueh that for aLL x,Y . A, o(x,y) = t¡(y,x)

ftu) g6 h¡ith &iscv'ete topoLogy) ís type I.

To prove this theorem we introduce the notion of a

twisted group a'lgebra. For a discrete group G with

normal.ized mult'ipliêF rrr, the twisted group algebra A(G,r.,r) of

G consists of finíte'ly supported measures on G with multipf ica'Lìon.

(uv)x = Iy 6uyvr-i, (v-l,X),

and *-operat'ion

(u*)* = (u*-r)-,

u,V e A(G,ur), X,Y e G, where ux (x e G) denotes the measure of

the set {x}. Each r¡-1-representatioh n of G extends natura'lly

to a representation

u * Ig.G ugn(g)

of A(G,r¡) which, when no confus'ion may arjse' we shalì denote by

the same letter. Note that the maP

A(G,ur) + V(G,o) : u + p(u) = IgeOuno(9),

97.

vlhere p denotes the regular o-t-representation of G, is a

*-monomorphism with range {a e V(G,u:) : supp(a) is finitei.

In parti cul ar, (A(G,ur) ) 'is weakìy dense i n V(G,t¡) .

PR00F 0F THEOREM 4.5. Suppose (i)'is true then by the

proof Theorem 4.4, there exists a group K such that

tG : Kl ( æ: K'= Ço and ,lf*f is trivial. Since the

maximal type I central projectjon of V(G,ur) is the'identity,

by Theorem 4.4,

167 IgeG v(g)p (g)1[=

hence by the un'iqueness (1.4.(ii)I è = {e} and thus K is

abelian. From r,rlç*ç trìvial , we can novl conclude ihat

o(x,y) = r(y,x) all x,y e K. This proves (i) implies (iii).

Suppose (iii) is true. Let H be a subgroup as describeC

in (iij), and r ôh o-I-representat'ion of G. If gr, ... , gn

is a set of coset represent.atjves modulo H, then

A(G,o) = A(H,ru)p(gr) ' ... @ A(H,t")o(Sn)

where the direct sum is a module direct sum. Again, by

representing A(G,t¡) by right multipl'ication on itself , we

can embed A(G,o) in the n x n matrices over A(H,o). By

hypothesis, A(H,o) is abelian and so A(G,o) satisfies a

polynomiaì identity (I.4.2). Sìnce the von-Neumann algebra

9B

generated by n ìs the weak closure of n(A(G,r)) by I.4.5 and

I.4.6, n ìs type I. This proves (iii) impìies (iì).

The implìcation (ii) implies (i) is trjvial.

Suppose we have [G : A] < - ând ,(x,y) = ,(y,x) al l

X,) e A, then [G': A'l < - (4.1 (ii¡) and Ao is abelian,

so by the equivalence of (i) and (ii), Go is type I.

Conversely, if Go is type I, then an abelian subgroup of

finite jndex in Go projects onto an abelian subgroup of

finite index in G on which ur ìs symmetric, so (G,ur) is

type I . Thus (i i i I and ('i v) are equi val ent.

COROL|..ARY 4.6 . The subgroup A in 4.5 ('ii i) may

be taken to be Zr(l).

Thus

tG

PR00F_. G' type I implies [G' : Z((G')fC)] . - (I.7.10).

lG : zr(n)l = [G' : (zr(^))'] = [c' : Z(n0¡1 =

I z((G )rc)l < æ usìng Proposition 4.1.

COROILARY 4.7 ([3, Lemma 3.1]), If G is a discrete

abeLian group, then V(G,r) .is type î if and on,Ly íf

Z'r(G) = Gr¡ = {g . c : o(g,x) = r(xrg) = I aLL x < G} has

finite ínã.eæ in G.

PR00F. The ''i f ' part fo'l I ows trivi aì ly from Theorem

4.5. Suppose V(G,or) ìs t¡rpe I, then for each x e G, Cr(x)

contains the group 7u,(t) which according to Theorem 4.5 has

99

fin'ite index in G, thus ¡ = G and [c : Zr(G) ] . -.

COROLLARY 4.8. If G is a discrete abeLian .Çroup, then

either V(G,oJ is þape I oz, Y (G,o) is type tt.l,

PR00F. Suppcse the maximal type I part of V(G,r) is

non-zero, then by Theoren 4.4, there exists a subgroup H

such that [G : H] < * ând ,lH*H is trivìal . S'ince H 'is

automatically abelian by Theorem 4.5, V(G,r) is type I.

Theorem 4.4 is somewhat awkward for practical use

because 'it is generally dìfficult to establish if a

group and multipl'ier satisfy cond'itions 4.4(b) or 4.+(c).

One would like to replace these conditions by [G: ^] ( -,

l¡'l < - ând rn it trivial for some n. The foilow'ing

examp'le shows this cannot be done.

EXAMPLE 4.9. Let G be the discrete group H x H',

where H = JI - 7(2) and H, = (D .- Z(2) and defineJ=j=1

,((aj,bj),(ar,,b¡')) = exp t}l I;=i(ajbj' - uj',br')1,

(aj,bj),(aj,brj) e c. By 4.7 and 4.8, V(.G,o) is type n1,

despite tlre fact that G and o sat'isfy tG : Al ( -r

l¡'l < @ and o2 = 1.

EXAMPLT 4.10, Let G = 7 x Z and

o((m,n)(mln')) = exp [2nicr(mn' - m'n)].

10(

Here V(G,r) is type I if cr is rational and type II1 if cr

is 'irratinal . (Use 4.7 and 4.8).

EXAMPLE 4.11. Let p be a prìrne and let G be the

group wi th fi ni tely many generators b,â 1 ,â2 . . . . and the

definjng relations

atb = bat ì = Ir2,

bP=arP=uf

ui*k ai = baiai+ft ì,k = 'l,2,

ti, tl un'n' ) = exp I+ ( sr så - sïs2l )Sn

The elements of the form bs'a,

It is clear that the commutator subgroup of G coincìdes

w'ith the centre cf G and is equa'l to the finite cyc'lic group

<b>. This itnp'ì'ies that there is a bound on the size of the

conjugacy classes of G.

sb a1 un bo

I S sô4 a n form a normal

n

subgroup H of G that is contained in A, hence [G: A] < æ;

also lA'l = lG'i = p, so by Theorem 4.4, V(G,r) has a non'zeno

maximal type I part. However, because Z(¡) n H g Z(H) = <b>,

the group Z(n) and consequently Zr(a) has infinite index in G,

thus by Theorem 4.5, V(G,r) is not type I. l^Je have shown that

Corol'lary 4.8 does not necessarììy hold for non-abelian groups.

Let o be the mult'iplier

S" i(

4

10

The foliowing theorem is an appl'ication of 4.5 to jnduced

characters.

THEOREM 4 12. Let G be a díscz,ete group and N a normaL

stbgroup of G, If ), Ì.s a aharacten of N, that ís a I-dimensíonal

nepz.esentatíon of N, d.enote bU Kx = {g € G : gng-i¡-re ker }.

aLL n e Nj the stabilì,zen of À., then the üon Newnann algebra'

Y

^ genez'ated. bg the induced z'epz'esentatíon ),+f;, is t14pe I

íf, and onLy if K^ contaíns a subgroup A sueh that [K¡. t A] < æ

ma Ar c ker À.

PR00l. Let P c K^ be a set of coset representatì ves

modulo N (includ'ing the identity of K^). For each x e K^ denote

bV sx the unique element in P such that xN = o*N. Let rf be the

functionG+[rl ; x+cr;rX. The relat'ionsû(n) =n, r](xn) =ü(x)n

and ú(nx) = {,(x)x-lnx, for x e H, n e N are evident. It js clear

that i : x -> r(V(x)) is an extension of À to KÀ and that the

multiplìer o(x,y) = l(ú(xV))r(,p(x),t,(V))-1 assoc'iated with thÍs

extension can be factored through K./N to yield a multìplìer ur

on K^/N. We may assume without loss of generality that o'is

normalized. By [24, Theorem 7], the von Neumann algebra V^

generated by the induced representation ^^fi

ls *-isomorphic

to V(K^/N,o), thus by 4.5 we deduce that V^ ìs type I if, and

only if there exists a subgroup A in K^ such that both ('i)

[Kr : A] < - ând (ìi) A' g N, o(x,y) = Õ(y,x) aLL X,Y e A,

hold. It remains t,o show that (ii) 'is equivalent to A'c ker ^.

Suppose (ii) is true. Let x,y e A. S'ince x-ty-lxy e I'1,

u(xv) = q(yxx-ly-lxy) =,/,(yx)x-Iy-1xy, but from o(x,y) = o(y,x)

follows r(V(yx)) = r(q,(xy)) so that À(x-tt-rxy) = 1. Conversely

ìf A'c ker À c N, then fon x,y e A,

10Í

r(xy) = r(u(yxx-rv-rxv)) = r(.p(vx)) À(x-tt-1xv) = r(u(vx))

and thus o(x,y) = o(y,x). Thìs completes the proof.

103.

CHAPTER iV2

GROUPS WITH FINITE DIMENSIONAL IRRTDUCIBLE MULTIPLIER RTPRESENTATIONS

Let G be a localìy compact group and u a normalized

multiplier on G. Recall that V(G) denotes the von Neumann

aìgebra generated by the ìeft regular (ordinary) representation

of G (1.7). In Chapter III we defined and. investigated the

structure of the von Neumann algebra V(G,or) in the case when

G is discrete. In the more generaì situation when G'is ìoca'l'ly

compact (but n,ot necessarily discrete), the definjtions are

sjmìlar. Indeed, the map p : G + U(12(G)), defìned by

p(g)f(x) = o(g-I,x)f(g-1x)

a'lmost everywhere in x, all g e G and f e L2(G), is a

t,-I-representatjon of G and generates a von Neumann algebra

also denoted by V(G,t¡). p 'is called the ìeft regular o-l-

representation of G.

In th'is chapter, we determine necessary and suff ic'ient

cond'itions on G such that the maximal type I¡ centraì

projection in V(G,ur) js non-zero (respect'ively the ident'ity

operator in V(G,r)), and construct this proiection expl'icitly

as a convolution operator on L2(G). This extends the results

of Chapter III

2After finishi.ng this thesis, I received a preprint entitled'rThe type structure of multiplier repiesentations rvhich vanishat Infinity?r by E. Kaniuth and G. Schlichting, which containsresults that are similar to those presented in this chapter.

10

For the case of ordinary representations, th'is probìem

has been successfully dealt with by Kamith [23] and Tayìor [48]

We point out that it'is no longer the case as'it was

for G discrete (see III.1.6(iv) and I.3.2) that V(G,r) is

necessarily finite. For an examp'le see 4.1. It is for this

reason that our methods allow a characterìzatjcln of the type

I, part, but not the type I part in V(G,o).

Recall that for a locally compact group G, Go denotes

the von Neumann kernel

G ñ{ker n : n e G^ and dinl n < -}.

and G* denotes the topological finite class group of G

wh'ich consists of all elements of G that belong to a relatively

compact conjugacy class (I.7). If ur is a normalized Borel

multiplier on G, then Go denotes the central extens'ion of

G (I.6). For a subset ll of G, H- denotes the closure of H in

G, and if H is also a subgroup, H' denotes the commutator

subgroup of H.

1. Prel 'imi nari es

Throughout this section, G will denote a locally compact

group and o a Borel multip'lier on G.

o

1ûl

LEI'4MA 1 . 1. Let A be a subset of Gu and. Let h : G0 -,' G

denote the eanonicaL proiection, then

(í) A- is cornpaet if anÅ onLy íf h(A)- is

contpact,

(i¿) ¿f A- is conrpact, then h(A)- = h(A-).

PR00F. A- compact 'impl ies h(A-) compact, but h(A-) ) h(A)'

hence h(A)- is compact and h(A-): h(A)-. Conversely, if h(A)-

'is compact, then by [18, 5.24], h-I(h(A)-) is cornpact; but

h-1(h(A)-) r h-lh(A) ) A, hence A- is compact and

h-1(h(A)-) ) A-, that is h(A)- = hh-1(h(A)-): rr(A)-. This

completes the proof.

C0ROL|-ARY 1.2. Let G be a LocaLLy conrpact gt'oup and u

a norTnaLized Borel muLtipLíer',

(¿)

(ii)(GFC)' = (c')rc.

rf one of (G¡rç) '- arld ((G')FC) - is cornpact,

then so is the ot'her and (Gp) ' = ht((G')FC)'-1.

PR00F. For (i), let A equal the coniugacy class of some

(t,x) e Go and in ('ii), let A = ((Gur) FC)'. Now use Lemma 1.1.

Combinìng 1,2 wtth I.7.8 shows that the maximal type

I, central proiect'ion'in V(G) is non-zero jf and on'ly if the

maximal type I, centra'l pnoject'ion in V(Go) i s non-zero. Thi s

follows also from Tay'lor [48, Proposition 5.2].

106

Let En, n e v-be the maps on L2(G') g'iven by

tnf (t,x) = tnfnr-nt{s,*) ds,

for alrnost all (t,x) . Gt, f e L2(G'). (Compare this with

K]eppner 124, page 5631.) Cìærty EnL2(Gur) consists of all

those functions f. t2(G') such that f(t,x) = tnf(1,x) for

almost all (t,x) e Go. It follows that the En, n e Zare

mutua'lly orthogonal idempotents. For n e 7-, En is just

convolution by the measure obtained if you multipìy the

measure on Go supported on 'fwhjch restricts to Haar measure

on 11, w'ith tne character fn of T'g'iven Oy fn(t) = th, t e T.

It is easy to check that En commutes w'ith the right and left

regu'lar representation of Go, so by I.7.4, tn is in the centre

of V(G0).

THEOREM 1.3. With the aboue notatíon' tVÌ.e E-n, rl € 7.

are mutuaLLy orthogonaL centz'al proieetions in V(Gu) md

the tht ee pon Netnnann aLgebr'as En V ( Gu') , V (G , on ) ond(i)

ft¿)

nülV(X_n+ff. ) - *e Don Neumavtn aLgebz'a generated by

the induced. z,epresentatíon Ï-n*$I - dve spatiaLLy

ísomorphic,

In.Z En = I (the identity operator).

PR00F. (i) Let t denote the left regular representation

of Go and on, n € V-, the left reguiar on-representatir¡n of G.

Observe that the representation space or X-ntfl is .just EnL2(Go)

10

and that Enr - .$'. It follows that Entl(Go) = v(u-n+$')./\ n

Denote by i the injection map i : G + Go and by q the

map

0:E L2(G)+12(G):foin

(Inle insist that Haar measure on Go be scaled so that the

measure of'ìf Ís 1.) Since Haar measure on Go is the product

of Haar measure on G and Haar measure on'ìi', we ha.ve

I r(t,x) | 2d(r,x) = I, [I" t,t'x)l' o'] dx

GüJ

I o(r) (x) | 2 dx,

G

thus q is an 'isometry. The spatial monomorphism

En\/(co) *g(t-2(e)) :T+0o JoO-t

is weakìy contÌnuous and maps Enr(1,x) to pn(x). Since

tnEnr(t,x)= Ën.(t,x), the von Neumann algebra generated by

{Ent(r,x) : x € G} is precisely EnV(G'). Thus any element

in EnV(Go) 'is the limit of operators T with the property

O(T) e V(G,orì), thus by the weak continuity of ö, O(T) e V(G,rn).

In other words, the range of O is a subset of V(G,rn). By

Sakai [40,1 .1.6 "21, the range of q ì s weakly cl osecl; i t contai ns

the operators 4 o En.(t,x) o ó-l = on(x), X e G and therefore

must equal V(G,rr).

10

(ii) From the theory of characters, the direct sum

nnr. Zn is the regular representation of T,

hence by the non-separable version of I.8,2,

we see that the induced representation

(* n.z.xn ) +et

is the regu'lar representat'ion of Go. Since inducing commutes

with taking direct sums (non-separable versÍon if I.8"3), we

have

*¡.7(xnrfi') = (* nuzxnl ^,f;.'.

Thus v(e0) = *nuv.v(xn+ff),

which is the required result.

LEMMA 1.4. ( r3l ). Let n be a u-z,epnesenta'bion of the

Dernte by p-ker r the eLosed. nonnaLLocaLLy compacl; group G,

subgnoup,

{x e G : n(x) = y(x) ,I, foz. some y(x) e 'llJ

of G. Then u is sirníLay to a rn¿LtipLierLífted from G/p-ker n.

PB00F. Let { be the canonical homomorphism from the

unitary group U(Hr), where H,,, denotes the Hjlbert space

associated with n, to the quotíent t-r(Hr)/T. According to Feldman

and Greenleaf [9], this map has Borel transr¡ersal, that is a

Borel map o, back from U(HîT)/11 to U(Hn), taking the identity

10

element ìn U(Hn)/'lf to the identity element in U(Hrr) such that

rf o a is the identity map. From this we obtain the commutative

di agram

e Iu(Hn) gu(Hr)/1t

,þi+dIt

u(Hn)

where n' is the map ü o {., o n. C'learìy n' is a multip'lìer

representation of G associated with a multiplier ur, Sôy, and

since rp " n = ! o Trt, (,r' is similar to t¡. From the definition

of n'' it follows that o' is constant on p-ker îT cosets in G x G,

that is, Ít'is ljfted from a multjplier on G/p-ker n.

LEMMA 1.5. Let G be a LocaLLy conrpact gnoup ui'bh

norrnaLized BoreL rru.LtipLer u. Suppose the maæinnL type I,centraL pz,ojection ín Y(G) (or equiuaLentLy in y(eo)) ¿s

non-zero, and that G admits a finite dimensionaL u-r,epz,esentation

lrr then K = h((G')") is conrpacb (uhev,e h is the canonicaL pz,ojeetíon

Go * G) an-d. u is simiLar to a nuLtipLíen uhich ds Lifted fnom

G/K.

Pß00F. That K is compact follows from 1.7.9. Suppose (t,x)

then since (s,y) -t sn(y), (s,y) . G', 'is a finite dimensional

(ordinary) representation of Go, we have I = tn(x), that isX e p-ker n. The result now follows from Lenlma 1.4.

. (G')" ,

1L,

2. The main theorems

THTOREM 2. 1. Let G be a LocaLLy cornpcct group uith norrnaLized

BoreL rruLtipLier u.

equiualent.

Then the foLLouing three conditions are

(i) The maæimaL type I, centz,al pz,oiection e ín

V(G) (oz, equiuaLentLy" the manimaL type I,

centraL pz,ojection in V (G')) is non-zey,o and. thene

eæists a firn te &imensí.onal. u-z.epz.esentation t

ol v.

(ii) Tke ma.ßnimaL tgpe I, centraL pz,ojection d in

V(G,ur) is non-zero.

(i'ùi) tG : Gpçl ( @: (GfC)'- ís conrpact and G adnits a

fì,nite dímensionaL ct-r,epresentation n .

PR00F. (i) and (iii) are obvjously equivalent by I.7.8.

Suppose (ii) is true and ler n be a finite dimensional representation

of dV(G,r). If we compose representation wjth the projection

V(G,o) + dV(G,o) : a + da, we obtain a representatioh r of V(G,o).

By I.6.3, the left regul êF o- l-representat'iot1 p of G corresponds

to a representation p'of Ll(G,,u-t¡ which v¡e know to be faithful

( [2,t9.3.6]). Thus p' is a *-monomorphism of Ll(G,r-1) to v(G,o)

and n o p' is a representation of ¡t(6,r-i). Again by I.6,3, this

representation .ôrr.sponds to a ,-l-representation of G and it is

easy to check that this o-l-representation is just g * n(dp(g)),

g e G. Hence g + n*(dp(g-l)), where * denotes the Hilbert space

adjoint, is a finite dimensional ür-representation of G. 0n the

other hand, by Theore 1.3, the max'imal type I, central projection

11

'in V(Go) is non-zero. This shows (ii) implies (i). That

(i) imp]'ies (ii) will be proved together with Theorem 2.3.

LEMMA 2.2 . Suppose K is a compact norrnaL subgz'oup of G

mtd u is Lif'bed fz,om a rruLtipLíez' u' on G/K, then K is also

a eornpaet norrnaL subgz,oup of G' and Gu/K is topoLogieaLLy

isomoz,phic to (G/K)"t.

PROOF. Let rf be the mforph'ism Go + (G/K)o' , (t,x) * (t,xK).

The kernel of this map is {(1,k) : k e K}, that is the image of K

in Go, thus the induced map from GolK to (G/K)' is an iso-

morphism. This isomorphism preserves the Haar measures on these

groups; they uniqueìy define the topoiogjes on these groups,

thus Go/K and (G/K)'' uru also topologically isomorphic.

THEOREM 2. 3. Suppose G is a LocaLLy eonrpacb gnoup uith

BoreL rru.LtipLier u. Sttppose the maæimaL type It centraL

pr.ojeetion d ín V(G,c.r) is non-zero. l,/e asswne (usíng 7.5 and

2"1) that u ¿s tifted fz,om a muLtípLier u' of G/K, uhere K ís

the eonrpaet nonnaL subgroup K = h((G') ") (h being the canonieaL

pnojeetion Go + G) , Ihten d is the opez,ator L2(G) + L2(G)

defdned by

f(k-r¡)dÀ(k),K

aLmost aLL x e G, f e L2(G), uhere À. is Haar measu.re on G

normaLízed sucTt that x(K) = l. Fuztherrnone, for eaeh n e Z, d

df(x¡ =

ís the ma,æimaL type I, centz,aL pnoiection in Y(G,rn) and dv(G,r¡n)

111

is isomozphic to V(G/K, (r')n).

PR00F. First we give a proof, as promised, of the statement

'(i) impf ies (ii)' of Theoren 2.I. Let cr : L2(G) * Lz(G) be

defi ned by

af(x) = f(k-1x) dr(k),K

almost all x e G, f e L2(G). The proof that a is a central

idempotent in V(G,rn) = EnV(G') (and hence in V(G')), and that

qV(G,on) and V(G/K6')n) are spatially isomorphic is sjmilar to

the proof of t'he correspond'ing facts about En in Theorem 1.3.

Si nce

v(G,rn) - v(G/K,(r')n) * v(e/r,(r')n)t

a^(where r denotesþrthogona'l

1.3'

q.Lbtlqe'V.<¿L

€ome+èñefi,i) , lve have by Theorem

V(c') = V((c/K)'') * V((G/Kt")r

but (G/K)'' and G'/(G'). are topologically isomorphtc (2.2) and

V((G')/(G'). ) is spâtjally isomorphic lo the maximal type I,direct summand of V(Go) (I.7.9). In particular we have d = a I 0

Now assume d I 0, then by (ii1 implies (i)r of Theorem 2.I,

the maximal type I¡ central projection in V(G') is non*zero, thus

by the same argument as above, we reach the desired conclusion.

1i:

COROLLARY 2.4. Suppose the manimaL type It centraL

projeetion d in V(G,o) ís non-zero, and Let n e 7-, then the

foLLouing equations obtain,

G" = h [((Gt)"] = h [n {ker r : r is a finite dimensiortnL

representation of G^ such that

nlrr(t) = tnÌl

' {g e G : there erists y(g) e Tl such that

,r(g) = y(g)I for aLL finite dimensionaL ,î-

repnesentatíons of Gj, .

uhez,e h : et * g denotes the eanonical pz'ojection.

PR00F. Let K = h[(G').] and denote the last two sets in

the above equality by H and L respectively. That Kc- H c L

is clear from the definit.ions and the property that a on-

representation n of G extends to an ordinary representatìon îT' of Go

such that n'l'(t)= tn. Using the proof of Lemma 1.5, we assume that

t¡ i s I j fted from a mul t'ipl i er on G/1. The fi ni te dimensi onal

representations of dV(G,rn) separate the points of dV(G,rn), hence

n'(9) = 1 if and only if t'*(dp(g-1)) = r*(d) for all such

representations nT where no denotes the ürn-representation g * n*(dp(g-1))

and p is the regular urn-representation of G. Thjs happens if and

only if p(g)d = d. If g e K, then c'lear'ly p(g)d = d. Conversely,

suppose g I K and p(g)d = d. Let U be a neighbourhood of gK'in

G/K of finjte Haar measure, not containing K. Let rp' be the

characteristic function of U and ,þ Y^lifting to G, then rl e dlz(e ),

is continuous at g and satisfies V(gK) = {1}, ü(K) = {0}. Since

ll¿

p(g)d = p(g), we have

,(g-1'x),¡l(g-1x) = ú(x) ,

almost all X e G. Substitut'ing g = x int<l this formula gjves

Q = lV(t)l = lV(g)l = 1 which is a contradiction. tlle have shown

that n"(g) = I if and only if g. K, thus

K = n {ker n' : n is a finite dimensional

representation of dV(G,rn) ]

)L

If we let n = 0, we obtain the remainder of the corollary,

thatisG=1.

LEMMA 2.5. (Moore [31], Lemma 4.1). Let G be a LoeaLLy

Conrpact gï.oup uith norTnaLízed BoreL muLtipLiez' u, Each

iw.edueibLe u-r'epresentation of G is finite dime'nsionaL íf and

onLy if euevA u-Tepresentation of G ís finíte,

PR00f. The if part is clear because an irreducible

to-FêpFêsentation is finite if and oniy'if it is finite dimensÍonal.

Let A be the twisted group C*-algebra C*(G,o) (see I.6). By

TheorsnI.6.3,'it is sufficient to prove the corresponding statement

of the Lemma for representations of A. If I is a two sided primitive

ideal of A, then I is the kernel of an irreducible representation of

A. According to our hypothesis, n has dimension n for some n.

Since r¡ ìnduces a homomorphism for A/I to the n x n matrices, A/I

satisfies the polynomia'l identitY Szn (I.4). Let Fn be the set

i1i

of primitive ideals I such that A/I satisfies the polynomial

identity Szn, then by assumption F = UFn is the primitive'ideal

space of A. For each subset K of F, the kernel I(K) of K is

n I (I € K). The closure of K in the kerne'l-hull topo'logy on

F is K- = {J e F: J r I(K)}. hle show that Fn is closed in

this kernel-hull topo'logy. Sjnce S2n(A) c I for each I e Fn,

the polynomial Srn 'is satisfied in Al I(Fn). Moreover, ifJ e Fi then Srn(A) c. I(rn) c,l, hence J e Fn. Thus Fn is closed.

t^le define for each closed subset K of F and for every

representation n of A, Pn(K) to be ihe proiection onto

H-(K) = {x e H-: n(a)x = 0 all a e I(K)}. According to1I' TI

[16 : Theorem 1.9:l , f -' Pn(K) extends to a countably addjt'ive

projection valued measure on the Borel set,s of F with P,r(K) in

the centre of the von Neumann algebra V(n) generated by n. Since

F = UFn, Hn = I(H,r(Fn) - Hn(Fn-r)). We define a subrepresentation

nn of n by restrictihg n to the invariant subspace Hn(Fn) - Hn(Fn-r).

Then n = Si nce S 'is satisfied211

in A/I(Fn) as noted above, the algebra nn(A) and hence its weak

closure V(nn) satjsfjes Srn = 0. Now any von Neumann algebra

satisfying this identity is finite (I.4.5). Moreover V(r) is the

direct sum of the V(nn) and since the direcl sum of finite a'lgebra

is finite (I.3), V(n) is finite as desired.

Let G be a LocaLLy eorrpact group and u a norvnalized

[nn, and n,.'(a) = 0'if a e I(Fn)"

BoreL rm,útipLier on G" The foLLouing are equíuaLent:

THTOREM 2.6.

(¿) V(G,tr) ís t11pe If.

11r

ø¿) ALL irz,educible u-nepresentation of G are fínitedimensíonaL.

(iii) The foLLouirry conditions hoLd.

(d [G : GrC] ( *,(b) (GrC)'- is cornpact,

(e) G adniis a finite &imensionaL u-nepresentatíon,

(il G = {e}.o

(iu) Gu ís a \koore gr¿oup, that is aLL its iz'reduc.ible

(ordLnany) repnesentations are fíníte dimensionnL.

PR00l. (i) 'implies (iii). By Theorem 2.1, tG: GpçJ ( @s

(GfC)'- is conípact and G admits a finite d'imensional o-representation.

Further by 2.3 and 2.4, V(G,tr) type I, implies

{e}=K=h((G')o)=Go

(iii) impì'ies (i). By 2.I, the maximal type I, central

project'ion in V(G,ur) ìs non-zero, so using 2.3 and 2.4, we obtain

V(G,ur) is type I.

(ii) implies (iii). Let A = C*(G,o) be the twisted group

C*-algebra of G. Since each representation of A is finite

dimensional, n(A) is contained in the compact operators for each

irreducible representatioh n of A, A is CCR (or lim'inal), thus

by Dixmier [7,5,5.27, A is type I, consequently V(G,r) is type I.

To see that V(G,o) is finite, apply Lenma 2.5.

(iii ) impìies (iv). If n 'is a finite dimensional o-representatjon

of G, then the n-fold tensor product n o ... @n is a finite djmens'ional

1i

,n-r.pr.rentation of G, hence V(G,rn) Ís type I, for each n e Z-

(recall that we know (i) and (iij) to be equiva'lent) and by Theorem

I.3, V(Gt) is type Ir. It follows from 1.7.6 that Go is a Moore

group.

(iv) impf ies (ìi ). If n is an irreducible t¡-representation

of G, then (t,x) -' tn(x) being an (ordinary) representation of Go,

must be finite dimensional by assumption.

Before we proceed to examples,'we need one other result, a

result obtained using Theorem 4.5 of Chapter III' which generalizes

Moore [31, Theorem 1] and Tay'lor [48' Theorem 2]

THEOREM 2. 7. Let G be a LoeaLLy cornpact gttoup and i': a

noz,maLized Bov,eL tnuLtipliez' on G. The foLlouing are equiuaLe.nt.

(í) V(G,o) í.s type I.¡, fot'some íntegen k, that is non-zez'o

ma,æ¿maL type In centz'al pz'oiections occur in

V(G,o) only if n < k.

(i.¿) G has an open abeLían subgroup ll of finite inÅ.eæ

in G such that the restriction of u to H ís triuiaL.

(íii) The ír'redacibLe u-r'epresentations of G are of

dimension at most k for some ittteger k.

PR00F. (i) impfies (iii). tle follow the proof of Taylor

[48, Theorem 2]. From I.4.5 we know that Sr_O = 0 is satisfied

in V(G,o). Suppose n is an jrreducib'le o-representation of G,

then r' : g + n*(g-t) is an irreducible o-t-representation of G.

Now, as jn the proof af 2.I, the left regular representation of

11i

L1(G,o-i) Ís a *-monomorphism into V(G,o). Hence L1(G,ür-1) also

satisfi.t srk. It follows that v(n') satisfies sr¡, consequently

?r' and r are of dimension at most k.

(iii) imp'lies (ii). Following the proof of Moore [31, Theorem

11, we consider the underlying group GO of G (with discrete

topology) and the corresponding twisted group algebra A = A(G¿,ur)

(see definjtion following III.4.5). To each r-1-representatiorì n of

G, there corresponds an algebra representat'ion no of

A : u * Ig.G uno(S). First we show that alì representatiohs no

obtained from irreducible o-I-representations r of G separate the

points of A.

The twjsted group algebra A acts on Lr(G,r-t1 as follows

u.f(x) = Igesupp uugf(g-1x)r(9-1,x),

almostall XeG, lreA, fe Ll(G,r-Ð. Givenanon-zerouinA,

it. is clear that u.f I O for some f e Ll(G,o-l). Let n' be an

irreducibìe representation of Ll(G,t,-1) such that n'(u.f) I 0 (I.5.3).

According to I.6.3, there exjsts an irreducible o-1-representation

rT of G such that

( n'(f)E,d ( n(x)E,.ù f(x) dxG

all Er¡ € Hn,f e Ll(g,ru-t¡. Using the invariance of Haar measure,

we obtain

ug I( rr'(u.f)t,nl =I

G

( n*t,rù ,(g-l,x)f (g-lx) dx

111

un In , ngrt,.ù r(g,x)f (x)

( nnn*E,.¡) f(x)f,

SrO(n(A)) = n(Sr¡(A)) = 0 for each suchn, that SrO = 0 is satisfied

I

I

dx

u dx

Hence 0 I "'(u.f) = no(u)"'(f). It follows that n'(u) I 0.

This shows that A has a separating family of representations

of degree at most k. I^le now infer, from the fact that

in A.

g

Let n be any irreducible t¡ ]-representation of GO and n' the

correspond'ing representation of A. By I.4.6 and I.S.1, Sr¡ is

satisfied in B(H?T) and we conclude that the dimension of n is at

most k. It now follows from III.4.5" that G has an abelian sub-

group K of finÍte index in G such that rlf*f is symmetric. The

closure H of K is an open abeì'ian group and if o = rlH*H, then the map

ã : H + H^ defined ¡V ã(g)(h) = o(h,g)/o(g,h) is continuous

(II.1.L), so it follows that o(g,h) = r(h,g) for all h,9 e H and

consequently (II.1.1), ,ittrH is trivial .

(ii ) irnp'lies (i ). Let 91,

representatives modulo H, then

, g¡ be a comPlete set of

A(G,ur) = *f=r A(H,r)p(g)

is a matrix algebra over the abelian algebra A(H,o), then as in

the proof of III.4.5, V(G,or) is tYpe I.O.

rzt,

4. EXAMPLES

Let G be a locally compact group wjth Borel multiplior t¡.

In Chapter III we saw that for G discrete, (G,o) is type I ifand only if GtD is type I and this occurs if and only if V(G,o)

is type I; and in this chapter, this has been generalized to

assert that for G locally compact, ihe following are equivalent.

(i ) G has only finite dimensional ur-Ì^êpresêntations, (ii ) Go is

a Moore group, and (iii) V(G,o) is type Ir.

Example II.3.4(iii) shows that (G,o) type I does not necessarily

imply that Go ,is type I and Mackey (129, Sect'ion 7l ) constructs a

non-type I group G such that V(c) is type I.

EXAMPLE 4.1. Let G be the group IR x IR and o the multiplier

r(x,y)(x',y')) = eioxy'

(x,y), (x',J') e lR X IR , r,¡here a is an irratjonal number. By

II.1.3, V(G,o) is type I. However, all irreducible ür-representatìons

of G are infinite dimensional [3, Theorem 3.3], thus by 2.1,

V(G,o) is not fjn'ite.

EXAMPLE 4.2. Let G and t¡ be as in III.4.9, then V(G,r) is type

Ir1, )êt G satisfies the first two conditions of 2,1(iii), thus G

admits no finite dimensional o-representations.

Now we expand on Exarnple 4.10 of Chapter III. For each

t = e2nio . 11, (o . lo,2¡r[), lve obtain a multiplier ort on L x 7-

defined by

T2

,r( (m,n) , (m' ,h' ) ) = t*n' ,

(m,n),(m',n') . Z x V. Theorems II.2..3 and II.3.5 show that

(up to similarity) all multipiier on Z- x V are of this form.

hle say that t is rational (respect'ively irrational) if q js

rational (respectively irrational). As'in Example IiI.4.10, we

see that V(V-xV,u¡) is type I if t is rational and type II1

if t is imational. Suppose o is rational , Sây a = p/q with

p and q relatively prime. hle wish to knovr more about the

trr-FêpFêSêntatjons of G and to this end, we util'ize Hannabuss

Il7,Theorem 4.1] which says that the irreducible or-representations

of G are all induced from characters of a maxìmal isotrop'ic sub-

group (tnat is a subgroup H such that Ho = H; see definition

following II.1.2). Now cìear'ly the subgroup H ='{(m,nq) : n,n e Z}

of G = ZxV- is maximal isotropic and G/H is isomorphic to Z(q)

thus the ot-representations of Z x V- are all of dimension q.

EXAMPLE 4.3" Let Ap (p a fixed prime) be the group of p-ad'ic

integers (for details see Appendix) and G the group VxZ. x ap

wi th mul t'i pì i cati on

(arbrx) (z' ,b' ,X') = (a + a',b + b',x + x' + ,r(ab')),

(a,b,x),(a',b',X') e G, where ú : L* op is the canonicai injection

of Z onto a dense subgroup of no (see Appendix). We topoligize G

so that lO becomes a compact open subgroup. With this topo'logy,

G becomes a I ocal ly compact separabl e topo'log'i ca1 group, For

each t e 'lf, we define a multiplier

12i

of = ûJt o k,

where k : G -> V- x V- is the canonical- homomorphism k(m,rì,X) = (m,n).

Given an irreducible ürt-representatiorì n of V x L, denote by

'rT' the or- representation of G obtained by composing ri with k.

Since nO'is compact, the abelian group dual op^ is discrete,

thus by I.8.4, every quasi-orbit on nO^ is transístive and all

the irreducible oa-representations of G can be constructed using

I.8.5, I.8.6 and i,8.7 .

Identífy,the abelian group dual nO^ of nO with the subgroup

^ = {s e 'lf : s = exp [Zrik/ pn],k ,t1 e Z] of 'lt using the cor-

respondence t\ x ap * T : (srx) -,' sx, where x * s*, s e  is the

continuous extension from Llo tO of the homomorphlsm V- -¡ Â :

n * ,n (see Appendix).

Let t e 'ïf and S e À, then the stabilizer of s in G is

all of G. Therefore all the or-representations of G are given

by I.8.7. Suppose we have a ot-representation y of G which

reduces to a multip'le of s on ÂO (recall that s is viewed as

a representation of nO). First we define an extension of s to

G as follows

: G+'ll': (a,b,x) *sx.s

The multiplier associated with this extensíon is preciseìy

72,

((a,b,x)(a' ,b' ,X')) * st a b xt b' xtS âr ,x S ,X'a ,

-5ab

6 ((a,b,x), (â',b',x' ) ).5

Thus by I.8.7, y must be of the form Tr'S', where r is an rts-t-

representat'ion of v- x v, (Note that orr-los = of as required. )

As s ranges through Â, we obtain all irreducjbìe oa representations

of G.

0bserve that ts-I is rational if and only if t is rational.

Hence the irreducible oa-representat'ions of G, t e T are all

infinite dimensional if t is irrational and are all fin'ite

dimensional if t is rational. It follows from 2.I and 2.6

that V(G,o¡) 'is type it if t is rational and the type If part

in V(G,oa) is zero if t is irrational.

Furthermore, if t is rational, as noted earlier, by varying

s, the ir"reducible urrr-t -representations of 7-x T. can be chosen

to be of arbitrary dimension.

The same is true of the oa-representations of G. Thus,

ai though V(G,r,r¿) is type I¡, Theorem 2.7 shows that V(G'tor) has

a non-zero maximal type In part for arbjtrary large n. This

phenomenon does not occur in the case where G is discrete

(combine III.4.5 and 2.7).

A.:

APPENDI X

Since there is no representation of the p-adic integers

ap, p-adic numbers nO and Q^ available for reference, which is

suitable for our purposes, we use this Appendix to set forth

such a representation, notation and some of the properties of

these groups.

1 no and co

The maps V(p'*n) * Z(pr) : x + x(mod pr) (p a fixed prime)

form an inverse system of (discrete) groups. Let Ap = Z(p"),

r = I,2, ; thus nO'is the closed subgroup of the compact

group tt;=r Z(pr) consisting of sequences (*n) such that

Xn = Xn+r(mod Rn).

lO is a topoìogical r'ing (under pointwise multiplication)

and each ideal prn' = {x + * x: pr-times, x e aO} is closed

(recal'l aO is compact) and is the kernel of the honromorphism

aO + Z(pr) , (xn) + x.i hence ptlp is open and the quotient

n'/prn' is isomorphic to Z(pr). In fact the groups prl' form

a neighbourhood base at 0.

The homeomorphism r/ : Z -> Lp: m + (rn(mo¿ pn)) is an injection.

Let x = (xn) . ^p.

Since ú(xr) - X e prlp, ìimn_r-ú(xn) = x;

ttrus ,/r(Z) ìs dense in lO. Simiìarìy pYZ is dense in prn'.

Furthermore, if (m,p) = 1, it is not hard to see that AO + lO :

x + mx is a continuous (and hence topoìogical) automorphism.

Note that the action of Z on lO is compatible with the ring

A

multipì'ication ìn aO, that is mx = x * ... * x = (m)x, in

parti cul ar pfx = ,lr(pr)x = (0, , 0, pFXt , pFxz ) (r zeros ) .

PROPOSITION 1.1. (Serre [43, II Proposition 2] ) . (i) For

æt eLement of A, (z,espeetíueLy Z(pr)) to be inuez,tib\e, it isneeessarA and suffícient that ít is not diuisbiLe ba p.

(¿i), If U denotes the gyoup of inuettíbLe eLements ofLrt euery eLement of Ap can be uritten tmiqueLy in the forrn

pnu uith u e U and. n > 0. (An eLement of lJ is caLLed, a unit).

PR00F_. It is sufficient to prove (i) for Z(pr); the case

of aO wil'l follorv. Now if x e Z(pY) does not belong tct pV,(pY),

its image in V(p) is not zeyo, thus invertible; hence there

exist !¡z e Z(pr) such that xy = 1 - pz, hence

xy(1 + pz + * pr-rzr'r) = 1,

whi ch proves that x 'i s i nverti bl e.

0n the other hand, if x = (xn) e aO is not zero, there

exists a largest integer r such that x. is zero; then x = pru

with u not divisible by p, hence u e U bV (i). The unjqueness

of the decomposition js clear.

PROPOS iTI ON .2 . Each eLosed stbgroup of Lp is aLso an ideaL

ín L, qnd aLL non-zerc íd.eaLs in L, are of the forrn pY AO for some

integer r > 0.

A

PL00t--. Let I be a non-zero ideal 'in nO. Since n prar = {0},

there exists a largest integer r such that prl' ¡ I; letx € I c prl

O be such that x . pr+1Á0, thus x = pru where u

is not cÍivisible by p (and is tlrerefore a unit). It follows

that pr1 - pruu-I e Iu-l g t, I f pra' and so I = ptlp.

Let l-l be a c'losed subgroup of aO. If x e H, then

nx = ú(n). x e H all o e Z-. Since lt(Z) is dense in n,, and

the ring multiplication in nO is continuous, yX € H ally . Ap, thus H is an ideal.

Next, we,define the field of p-adic numbers to be the field of

fractions of lo. Since elements x,x' 6 Ap are uniquely representab'le

in the form x = pmu,X' = pm'u', we have x/xr - orr-fi'(r(r')-I)),so one sees immediate'ly that nO = aO tp-11 and every non-zero

element of nO can be written uniquely in the form pnu with n e Z,

u e U. Addjtion and multipl ication in CIp are are follov,rs: given

two elements in oO, one can write them without loss of generalìty

in the form pmx, pt*sy, where s > 0, ffi e v, xry e tJ., their product

is p2ln+'s*, and their sum pm(x + psy). Endow no r^rith the topology

such that ÂO becomes an open subgroup. hlith this topoìogy CIp

becomes a I ocal ly conrpact topo'logi cal fi el d contai n'i ng Q as a

dense subfield. Finally, nO is metrizable, a.convenient metric

being d(pnt*,pny) = 2-tîin(m'n),rì1,n e Z,x,v € u; the restriction of

d to lO is also a metric.

pObserve that A and Qp are torsion free and nO is divisible.

A

Ap^ and CIp^

Embedd 7(p*) (respectively Z(pr)) the natural way onto a

subgroup ri, = {exp(Z"ik/Rr) : k,r e L} (respectìvely Âr) of '11".

Let  (respectiveìV Âr) have the discrete topology. For fixed

À e ft, the map Z- -> Iy : n + Àn extends to a continuous homomorphism

Âp * Â : À + rx defined as follows: if À e Â,^ ÀX = ÀXF, where

x = (xn). It is easy to check that ix is well defined.

Suppose x. ap ,xf l,thenbecause A ='{x e ÂO: lX(*) -11<6}

is an open set, there exists an integer r such that RraO gR. Itfollows trom álementary considerations that o(ptop) = {l}. Let

r be the smallest integer with this property. This gives rise to

the following commutative diagram of continous maps

2

X->op 1t

txt0

+ z(p")

where h is the canonical projection, o the isornorphism

(xn)Rrno + X" ând x' is necessarily of the form x'(a.) = Àa,

a e Z(pr) for some À € ^r

c ^.

It follows that x(x) = rx.

Converse'ley, for each À e Â,n C ¡, x * lX is a character of lO.

Summing up, we have up^ = ^,

A[Ap^,ptAp] = À" and by the

Pontryagin duality theorem [8,24.8] a pairing (À,x) = ¡X,

ÀeÂ,XeAO.

The method to find nO^ is simjlar; we claim that nO^ is

topologicalìy isomorphic to nO, the pairing being

lo/ptno

A

(pt*,pty) = exp lzni(xy)-r-s/pt*tl,

râs € T.,x,y e U. First we note that for each psx, the

map pry * (pt*,pry) is a (continuous) character of nO. To

show that each character is of this form, let x. ep be non-

zero. As before, we select the smallest jnteger r (possibly

negative) such that *(ptop) = {l}. For not'ional convenjence,

whenever pFx . ep, we deem xn = 0 for n < 0. The epimorphism

0 : np * n : psx + êXp[2nix.-rlpt-s] has kernel pt¡p. Again

we construct a commutative diagram of continuous maps

and using the previous correspondence n^ = Âp, we see that

x(psx) = [0 o h(psx)]V = (psx,p-ry) for some y e U 9 Ap.

Finally, it follows easiìy from the definition of nO^ that the

isomorphism 0p -, np : prx * (prx,.) is bicontinuous.

3. The dual of Q

+'lf

f

np

+h

n /p"ro I n

(

Let x e ú^. For positive integers n, let an =l(l/n!), then

= on_1,fì = 2,3, Conversely, given a sequencen

on)

{cn} c 'lf such that (on)n = on_1, n > 1, then x(m/n!) = (on)t

defines a character of Q ( [18,25" 5] ). Thus Q^ is the projective

limit of the groups Tn = lf, n = 1, ... ; the mappings being

tn + Tn+r : a. J. One of course has to check that the dual

topo'logy on Q^ agrees with the inverse limit topology of the Tn

A

Suppose {cln} e Q^ and onk = L, n = 1, ... Then for

n=!,2, ,1= (onoo)n=onkno=ono-r, thus or., = lalln, so Q^ ts torsion free. We divide elements in Q^ as

follows. Given tcn] . Q^, let ßn = on+m(nfit)(m fixed) where

(nf,t) (m fixed) wher. (nfrt) = (n + m) !/n!m!. The sequence

{ßn} is asain rn rnirrhar i, ßn = rÍl*t)n = on-1nn1Tm¡ = ßn-r.

Also (un)*' = on*rtñitt = on. hle have shown that if x t Q^

corresponds to {crn}, then À = {ßn} is the unique (because

is torsion free) element in Q^ such that im! = x.

L

2

I

9

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