abstract harmonic analysis on locally compact abelian groups1222456/fulltext02.pdf · regarding...

U.U.D.M. Project Report 2018:17

Examensarbete i matematik, 15 hpHandledare: Andreas StrömbergssonExaminator: Veronica Crispin QuinonezJuni 2018

Department of MathematicsUppsala University

Abstract Harmonic Analysis on Locally Compact Abelian Groups

Tobias Mattsson

ABSTRACT HARMONIC ANALYSIS ON LOCALLY

COMPACT ABELIAN GROUPS

TOBIAS MATTSSON

CONTENTS

1 Introduction 1

2 Topological Groups 2

3 The Measure algebra & The Group algebra 4

4 Representation Theory 5

4.1 Representations of the Group Algebra . . . . . . . . . . . . . 6

5 The Dual Space 8

5.1 The Fourier-Stieltjes Transform . . . . . . . . . . . . . . . . . 10

5.2 Functions of Positive Type . . . . . . . . . . . . . . . . . . . . 11

5.3 Bochner’s Theorem and the First Fourier Inversion Theorem 13

5.4 Plancherel Theorem . . . . . . . . . . . . . . . . . . . . . . . 15

6 Pontryagin Duality 16

7 The Culmination: Decomposition of Unitary Representa-tions 20

1 INTRODUCTION

The main purpose of this paper is to prove the decomposition theorem for uni-tary representations of locally compact abelian groups (abbreviated LCA),but we also intend to cover—what the author thinks are—the main resultsof abstract Harmonic analysis. The full corpus librorum can be found atthe foot of the document. We follow G. Folland’s book A course in abstractharmonic analysis throughout this paper.

Moreover, we assume that the reader is familiar with measure and integrationtheory, and functional analysis, in particular we assume the knowledge of the

1

2 ABSTRACT HARMONIC ANALYSIS ON LCA GROUPS

norm, strong, weak-* topologies and the topology of compact convergence,these topics are covered in any functional analysis book.

There are some results outside the scope of this article that we will come toneed in certain situations, e.g. various Riesz representation theorems, theGelfand-Raikov theorem, the Stone-Weierstrass theorem (locally compactspaces version), and the Banach-Alaoglu theorem.

2 TOPOLOGICAL GROUPS

The setting which concerns us in this paper will be that of the topologicalgroup, it is defined as follows. Suppose G is a group and a topological spacewith multiplication · : G × G → G and inversion g−1 : G → G, then G isa topological group if the inversion and multiplication operations are con-tinuous maps. Furthermore, a locally compact group is a topological groupwhose topology is locally compact and we implicitly assume the Hausdorffproperty. We follow Folland [1, Ch. 2] in this section.

Our most powerful tool will be a function called the Haar measure, it comesin two types, the left and the right variants, each giving the direction oftranslation invariance. More precisely, a left (resp. right) Haar measureµ is a nonzero Radon measure on G which satisfies µ(xE) = µ(E) (resp.µ(Ex) = µ(E)) for every Borel measurable set E ⊆ G and for every x ∈ G.We note that µ(E) is a left invariant Haar measure if and only if E 7→ µ(E−1)is a right invariant Haar measure, the proof of this is trivial, but it showsthat the theory is essentially equivalent in either case. We will be working inthe intersection of these cases, when the Haar measure is both left and righttranslation invariant, since our topological group will always be abelian. Andlastly, the Haar measure is called normalized if µ(G) = 1.

There is a relationship between left and right invariant measures called themodular function ∆ : G→ R ∈ Hom(G,R), which in some sense measures theextent to which a left (right) invariant Haar measure is right (left) invariant.When G is abelian ∆ ≡ 1 (in which case we also call G unimodular) sincethe Haar measure is both left and right invariant, thus it will not be relevantto us, but it is worth mentioning.

It is a fundamental—and incredible—fact that there exists an essentiallyunique Haar measure on every locally compact group G.

Theorem 2.1. Every locally compact group G has a Haar measure.

We refer to Folland [1, p.37] for a complete proof of this theorem, we willjust give a sketch of it here.

The idea is to utilize the Riesz representation theorem, so we want to con-struct a positive translation invariant functional T on Cc(G), this is doneas follows. We pick some function φ ∈ Cc(G) such that it is bounded by 1,equal to 1 on a small open subset of G and is supported on an slightly largeropen set U . We also pick a function f ∈ Cc(G) that is essentially constanton the translates of U , then f can be approximated by f ≈

∑aiLxiφ, such

that we get T (f) ≈∑aiT (φ). This approximation will become exact as we

shrink the support of φ to a point and by an appropriate normalization we

3 TOBIAS MATTSSON

can cancel the factor T (φ), so that T (f) is approximated by the sums∑ai.

Regarding uniqueness we will just prove the case for abelian groups, sincethis is our main focus, but note that it is true for general locally compactgroups.

Theorem 2.2. If µ and λ are two Haar measures on the locally compactabelian group G then there exists a constant c ∈ R such that µ = cλ.

Proof. Fix g ∈ Cc(G) with total integral 1 under λ, and define c by∫Gg(x−1) dµ = c.

Then for any f ∈ Cc(G) we have∫Gf dµ =

∫Gg(x) dλ(x)

∫Gf(y) dµ(y)

=

∫Gg(x) dλ(x)

∫Gf(xy) dµ(y)

=

∫G

∫Gg(x)f(xy) dµ(y)dλ(x)

which by the substitution (x, y) 7→ (uv−1, v) and by Fubini’s theorem be-comes

=

∫G

∫Gg(uv−1)f(u) dµ(v)dλ(uv−1)

=

∫Gg(uv−1) dµ(v)

∫Gf(u) dλ(u)

=

∫Gg(v−1) dµ(v)

∫Gf(u) dλ(u)

= c

∫Gf(u) dλ(u)

thus we’re done. �

We shall now present a few examples of different types of locally compactgroups. First we have the groups (R,+), (R,×), the first of which has thestandard Lebesgue measure dx, and the second has the average dx

|x| .

There is also the discrete abelian group (Z,+), along with all the finitegroups, these have the counting measure or the normalized counting measurein the finite case. Notice that the finite groups are both discrete and compact,there are many more cases when we release the condition on discreteness e.g.(T,×), whose Haar measure is the pullback of the Lebesgue measure dx from(R,+) by the function f(x) = e2πix. Since T is compact, the whole space hasfinite measure (this follows by regularity) thus we can normalize this measure

as well. If one does this one gets the measure d(x◦f−1)2π .

If one is familiar with the theory on smooth manifolds one might realizethat they are all locally compact (since they are locally homeomorphic to


Rn), thus one might suspect that there are smooth manifolds that have agroup structure with smooth group operations (in particular locally compactgroups). This structure is actually called a Lie group and has been studiedextensively. In fact GL(n,R) is a Lie group, the Haar measure on this groupis the measure | detT |−n

∏i<j dαij on n×n real invertible matrices T = (αij).

Moreover, the matrix groups SL(n,R), SL(n,C), U(n), SU(n), O(n), SO(n)are all lie groups.

3 THE MEASURE ALGEBRA & THE GROUP ALGEBRA

In this section we follow Folland [1, Ch. 2]. We define the measure algebra,denoted by M(G), as the set of all complex Radon measures on a locallycompact group G. This forms an associative Banach algebra under pointwiseaddition and convolution of measures over C. To define what convolution ofmeasures means let µ, ν ∈ M(G), then the following map is clearly seen tobe a linear functional on C0(G)

φ 7−→∫Gφ(xy) dµ(x)dν(y) (3.1)

which satisfies ∣∣∣ ∫Gφ(xy) dµ(x)dν(y)

∣∣∣ ≤ ‖φ‖∞‖µ‖‖ν‖where ‖µ‖ is the total variation of µ over G. It follows that (3.1) must begiven by some measure in M(G) which we denote by µ ∗ ν (because of theRiesz representation theorem), it is also clear that ‖µ ∗ ν‖ ≤ ‖µ‖‖ν‖.

The following calculation shows that convolution is associative. Let µ, ν, δ ∈M(G)∫Gφd((µ ∗ ν) ∗ δ) =

∫Gφ(xz) d(µ ∗ ν)(x)dδ(z) =

∫Gφ(xyz) dµ(x)dν(y)dδ(z)

=

∫Gφ(xy) dµ(x)d(ν ∗ δ)(y) =

∫Gφd(µ ∗ (ν ∗ δ)).

As our main focus is on locally compact abelian groups we note that M(G)is commutative if and only if G is abelian, one direction follows immediatelyfrom (3.1), for the other direction, let δx ∈M(G) be the point mass measureat the point x ∈ G, then∫

Gφ(uv) dδx(u)dδy(v) = φ(xy) =

∫Gφdδxy

thus δx ∗ δy = δy ∗ δx if and only if xy = yx, thus M(G) commutative meansthat G is abelian.

There is a very interesting subalgebra of M(G) that turns out to be a Banach*-algebra, it consists of all f(x) dx where dx is the Haar measure on G andf is some integrable function. This subalgebra is thus naturally isomorphicto L1(G) under the map f → f dx. The involution is defined by

f∗(x) = f(x−1)∆(x−1)

5 TOBIAS MATTSSON

(which reduces to f∗(x) = f(x−1), in the case of G being abelian) andconvolution is inherited from M(G). Note also that Fubini’s theorem impliesthat f ∗ g ∈ L1(G) when f, g ∈ L1(G) since ‖f ∗ g‖1 ≤ ‖f‖1‖g‖1 and it alsoimplies that

f ∗ g(x) =

∫Gf(y)g(y−1x) dy

is absolutely convergent for a.e. x ∈ G. It is an easy check that convolutionin L1(G) agrees with convolution in M(G) when we embed L1. Becausethis subalgebra is so important in Harmonic analysis we call it the L1 groupalgebra.1

4 REPRESENTATION THEORY

We will need to consider two types of representations, both deeply connected—as we shall see—on the one hand we have representations of topologicalgroups on Hilbert spaces and on the other hand we have representations ofC* Algebras, in particular, representations of the group algebra. We startwith the representations of topological groups, following Folland [1, Ch. 3].

In this text we will denote the space of bounded linear operators V →W byB(V,W ) and B(V ) = B(V, V ).

Definition 4.1. Let G be a LCA group, and let U(Hρ) denote the groupof unitary operators on the Hilbert space Hρ, then a strongly continuousunitary representation is a homomorphism ρ : G→ U(Hρ) from G to U(Hρ)such that g 7→ ρ(g)u is continuous for every u ∈ Hρ.2

We will also quickly cover some more definitions that will be necessary. First,two representations ρ : G → U(Hρ), ρ′ : G → U(Hρ′) are called unitarilyequivalent or intertwined if there is some T ∈ B(Hρ,Hρ′) such that Tρ = ρ′T ,such T are called equivariant maps.

Furthermore if there is some closed subspaceM⊂ Hρ such that ρ(g)M⊆Mfor all g ∈ G then we call M a ρ-invariant subspace of Hρ. And if M is aρ-invariant subspace then we define the subrepresentation of ρ as ρM : G→U(M), ρM(g) = ρ(g)|M.

If there exists a nontrivial ρ-invariant subspace M ⊆ Hρ then ρ is calledreducible, otherwise ρ is called irreducible.

Moreover, the dimension of a representation ρ is the dimension of Hρ.

If we have a family {ρα : G→ Hρα}α∈A of arbitrary unitary representationsof G then we can naturally define their direct-sum unitary representationρ = ⊕α∈Aρα : G→ ⊕α∈AHρα as

ρ(g)(∑α∈A

xα) =∑α∈A

ρα(g)xα

1Convolution on L1 actually extends to other Lp spaces, more information on this canbe found in Folland [1, p.52]

2It should be noted that the weak-* topology and the strong topology coincide on U(Hρ)(see Folland [1, p.68])


for∑

α∈A xα ∈ ⊕α∈AHρα (Note that⊕α∈AHρα stands here for a Hilbert spacedirect sum see Folland [1, p.254]). Clearly every ρα is a subrepresentation ofρ corresponding to the ρ-invariant subspace Hρα .

Now we are ready for some theory. AsHρ is a Hilbert space we have a definedinner-product, thus it is interesting to consider the orthogonal complementof ρ-invariant subspaces, in fact:

Proposition 4.1. Let M be a closed nontrivial subspace of Hρ, then M is

ρ-invariant iff M⊥ is. Furthermore, ρ = ρM ⊕ ρM⊥. 3

Proof. For u ∈M, v ∈M⊥ we have 〈ρ(g)u, v〉 = 〈u, ρ(g−1)v〉 = 0 so ρ(g)v ∈M⊥. The first part is proved since M = M⊥⊥ . And the second part istrivial as every x ∈ Hρ can be decomposed into x = u + v for some u, v asbefore, and ρ acts on these independently as the subrepresentations. �

Theorem 4.2 (Shur’s lemma). 1. A unitary representation ρ : G→ U(Hρ)is irreducible if and only if all the maps in B(Hρ) that commute with ρ(g)(∀g ∈G) are the scalar multiples of I.

2. Let ρ : G → U(Hρ), ρ′ : G → U(Hρ′) be two irreducible representations.If ρ and ρ′ are unitarily equivalent then any equivariant map T is a constantmultiple of a fixed isomorphism.

Proof. We start with (2). If T is equivariant then T ∗ is also equivariant,and TT ∗ is clearly a bounded self-adjoint operator. Any closed invariantsubspace of TT ∗ is ρ-invariant, thus is trivial, and by the spectral theoremTT ∗ is a constant multiple of the identity. This also holds for T ∗T , hence Tis a constant multiple of a unitary map, in particular it is invertible, or thezero map.

For the first part, take ρ = ρ′ in (2). For the other direction assume ρ isreducible. Then there is a closed ρ-invariant subspace M ⊆ Hρ, now let Pbe the orthogonal projection onto M and let v ∈M⊥, w ∈M, then

ρ(x)Pv =0 = Pρ(x)v

ρ(x)Pw =ρ(x)w = Pρ(x)w

by prop. 4.1. But then ρ(x)P = Pρ(x), and we’re done. �

Corollary 4.3. All irreducible unitary representations ρ : G→ U(Hρ) of anLCA group G are 1-dimensional.

Proof. All the ρ(x) commute, hence by Shur’s lemma ρ(x) = ξI for all x ∈ Gand some ξ ∈ C, hence all one dimensional subspaces are invariant, thereforedimHρ = 1. �

4.1 REPRESENTATIONS OF THE GROUP ALGEBRA

In this section we consider the group algebra L1(G), not as embedded inM(G) but as its own object. First we define representations of the groupalgebra:

3This is not true for non-unitary representations.

7 TOBIAS MATTSSON

Definition 4.4. A *-representation ρ : L1(G)→ B(Hρ) of the group algebrais a *-homomorphism. We call ρ nondegenerate if there is no v ∈ Hρ suchthat ρ(f)v = 0 for all f ∈ L1(G).

Remark 4.5. As we will always require nondegeneracy of our *-representationsin this text we will implicitly assume that they are defined as nondegenerate,and just refer to them as *-representations.

We will shortly present a result which says that we can identify the *-representations with the unitary representations of G by

ρ(f) =

∫Gρ(x)f(x) dx

this is defined to act on Hρ in the sense that

〈ρ(f)u, v〉 =

∫Gf(x)〈ρ(x)u, v〉 dx. (4.1)

Clearly |〈ρ(f)u, v〉| ≤ ‖f‖1‖u‖‖v‖ and since ρ(f) is evidently linear we haveρ(f) ∈ B(Hρ). But before we can present the result that gives this identifi-cation we must prove that ρ(f) is actually a *-representation.

Proposition 4.2. For every unitary representation ρ : G→ U(Hρ) the map

ρ(f) =

∫Gρ(x)f(x) dx

defines a *-representation on L1(G).

Proof. As we have stated before f → ρ(f) is clearly linear and bounded, soit suffices to show that it preserves involution, is a homomorphism and isnondegenerate. We start with involution:

ρ(f∗) =

∫Gρ(x)f(x−1)∆(x−1) dx =

∫Gf(x)ρ(x−1) dx

=

∫G

[f(x)ρ(x)]∗ dx = ρ(f)∗

And to show that its a homomorphism we just need to show that it preservesconvolution since we already noted that it was linear:

ρ(f ∗ g) =

∫G

∫Gf(x)g(x−1y)ρ(y) dxdy

=

∫G

∫Gf(x)g(y)ρ(xy) dxdy

=

∫G

∫Gf(x)g(y)ρ(x)ρ(y) dxdy = ρ(f)ρ(g).

These calculations are justified by equation (4.1), i.e. applying the operatorsto u ∈ Hρ and taking the inner product with v ∈ Hρ. The last step is to shownondegeneracy, so suppose u 6= 0 ∈ Hρ and pick a compact neighborhood Vof 1 in G such that

‖ρ(x)u− u‖ ≤ ‖u‖ for x ∈ V.


Now set f(x) = |V |−1χV (x) where χV is the indicator function of V , then

‖ρ(f)u− u‖ = |V |−1‖∫V

[ρ(x)u− u] dx‖ ≤ ‖u‖

hence ρ(f)u 6= 0. �

Theorem 4.6. Let ρ : L1(G) → Hρ be a nondegenerate *-representation.Then there is a unique unitary representation ρ : G→ U(Hρ) of G such that

ρ(f) =

∫Gρ(x)f(x) dx ∀(f ∈ L1(G)).

Proof. See Folland [1, p.74] for a proof of this. �

5 THE DUAL SPACE

Let G be a LCA group. We follow Folland [1, Ch. 4] in the remainingsections. Then every irreducible unitary representation ρ : G → U(Hρ) is1-dimensional by corollary 4.3, thus Hρ

∼= C and there exists ξ(x) ∈ C forevery x ∈ G such that ρ(x)(z) = ξ(x)z in particular |ξ(x)| = 1 by unitarity,ξ(x) varies continuously with x, and ρ is a homomorphism in x, so ξ(x)satisfies ξ(xy) = ξ(x)ξ(y). We call such maps ξ characters and the space ofcharacters is called the dual group, denoted by G. We shall denote ξ(x) by(x, ξ).

We will use the notation σ(A) when A is a Banach algebra to denote thespectrum of A i.e. the set of nonzero algebra homomorphisms from L1(G)into C.

Proposition 5.1. Every Φ ∈ σ(L1(G)) is given by integration against acharacter ξ ∈ G, i.e. by

Φ(f) =

∫G

(x, ξ)f(x) dx (f ∈ L1(G)). (5.1)

This gives a nondegenerate *-representation on L1(G).

Proof. First we note that proposition 4.2 says that (5.1) is a nondegenerate*-representation on L1(G), so we only need to prove that every multiplicativefunctional of L1(G) is given by integration against a character, this we proveas follows:

Every Φ ∈ (L1(G))∗ is given by integration against some φ ∈ L∞(G), firstwe will recover such a φ, then show that it is in G. Let f, g ∈ L1(G) suchthat Φ(f) 6= 0, then

φ(f)

∫Gφ(y)g(y) dy = Φ(f)Φ(g) = φ(f ∗ g) =

∫G

Φ(Lyf)g(y) dy

hence φ(x) = Φ(Lxf)Φ(f) , locally a.e., but taking this as our definition of φ(x),

we have everywhere continuity. We also see that φ(xy)Φ(f) = Φ(Lxyf) =Φ(LxLyf) = φ(x)φ(y)Φ(f), thus we also see that φ(xn) = φ(x)n but since|φ| ≤M is bounded, hence it must be that φ(x) ∈ T. And we’re done. �

9 TOBIAS MATTSSON

Corollary 5.1. The dual group G can be identified with σ(L1(G)) by (5.1).

Right now G has no more structure than being a group, we remedy this bygiving it the topology of compact convergence, i.e. the topology of uniformconvergence on compact sets.

Theorem 5.2. The dual group G is a locally compact abelian group underthe topology of uniform convergence on compact sets, which coincides withthe weak-* topology inherited from L∞(G).

Remark 5.3. We will not prove the fact that this topology coincides with theweak-* topology, but we note that it follows from the theory of functions ofpositive type, and can be found in Folland [1, p.89].

Proof. We have already showed that G is an abelian group, so it suffices toshow that the topology is locally compact and that the group operations arecontinuous, we begin with the latter.

The topology is the topology of compact convergence which means that wehave to show that for two sequences {ξn}, {ηn} of elements of G and any com-pact subset K ⊂ G such that ξn → ξ and ηn → η converges uniformly on Kthen ξnη

−1n → ξη−1 converges uniformly on K, but this follows immediately

by the (pointwise) triangle inequality:

|ξnη−1n − ξη−1| = |ξnη−1

n − ξη−1n + ξnη

−1 − ξnη−1|≤ |ξn(η−1

n − η−1)|+ |(ξn − ξ)η−1|= |ξn||(η−1

n − η−1)|+ |(ξn − ξ)||η−1|= |η−1

n − η−1|+ |ξn − ξ|= |ηn − η|+ |ξn − ξ|

This gives uniform convergence for ξnη−1n → ξη−1.

Now we show local compactness. By theorem 5.1 G∪{0} equals Hom(L1(G),C),which is a subset of the closed unit ball of L∞(G) and is weak-* closed andhence weak-* compact by the Banach-Alaoglu theorem, hence G is locallycompact. �

We now prove a result which gives us methods for decomposing the dualgroup G into a direct sum of cartesian product when G is cartesian productof other locally compact groups. First we define ⊕i∈IGi to be all (xi)i∈I ∈∏i∈I Gi such that all but finitely many xi = 1.

Theorem 5.4. Let {Gi}i∈I be LCA groups.

1. If |I| <∞ then∏i∈I Gi

∼=∏i∈I Gi.

2. Moreover, if Gi is compact for all i ∈ I then∏i∈I Gi = ⊕i∈IGi for

arbitrary |I|.

Proof. We prove them in numerical order. Note that the inner productis on C. Each ξ = (ξi)i∈I ∈

∏i∈I Gi defines a character on

∏i∈I Gi, by


〈(xi)i∈I , (ξi)i∈I〉 =∏i∈I〈xi, ξi〉, furthermore all characters in χ ∈

∏i∈I Gi

are of this form, where ξi is given by 〈xi, ξi〉 = 〈(1, . . . , 1, xi, 1, . . . , 1), χ〉.

Now assume all the Gi are compact and let |I| be arbitrary. In the same wayas before we get the characters so we only need to prove that all but finitelymany of ξi = 1. By continuity, there is a neighborhood U of 1 in

∏i∈I Gi

such that |(x, ξ) − 1| < 1 for x ∈ U . By definition U contains a set∏i∈I Ui

where Ui = Gi for all but finitely many i. In which case ξi(Ui) ⊂ ξ(U) henceξi(Ui) is a subgroup of T contained in {z ∈ T : |t − 1| < 1} hence equals 1hence ξi = 1. And we’re done. �

Considering the case when G is finite we see that:

Corollary 5.5. All finite abelian groups are self-dual.

Proof. By the fundamental theorem of finite abelian groups and (1) of the-orem 5.4 we only need to show that the cyclic groups Zn are self-dual. Therepresentation theory of finite groups (see Sagan [3]) says that all characters

of cyclic groups are of the form χ(k) = ekπin where k = 0, . . . , n− 1, and are

all 1-dimensional, it is obvious that Zn = {χk : k = 0, . . . , n − 1} which isisomorphic to Zn by χk 7−→ k, thus we’re done. �

The following result is an amazing fact about the dual group of a compactabelian group.

Proposition 5.2. Let G be a compact abelian group with normalized Haarmeasure. Then G is an orthonormal set in L2(G).

Proof. If ξ, η ∈ G then |ξ|2 = 1 so ‖ξ‖2 = 1 because the Haar measure isnormalized on G. If ξ 6= η then there is an x0 ∈ G such that (x0, ξη

−1) 6= 1and∫

ξη =

∫G

(x0, ξη−1) dx = (x0, ξη

−1)

∫G

(x−10 x, ξη−1) dx = (x0, ξη

−1)

∫ξη

which implies that 〈ξ, η〉L2 =∫ξη = 0. �

5.1 THE FOURIER-STIELTJES TRANSFORM

There is a more convenient homomorphism that identifies G with the spec-trum of L1(G) than the one we saw in section 5, its called the Fourier trans-form and is defined as

Ff(ξ) = f(ξ) =

∫G

(x, ξ)f(x) dx

where f ∈ L1(G), ξ ∈ G.

It is a notable fact that there is a version of the Riemann-Lebesgue lemmathat is valid in this general context. It is a part of the following proposition.

Proposition 5.3. The Fourier transform is a norm decreasing *-homomorphismwith a dense image in C0(G).

11 TOBIAS MATTSSON

Proof. The proof that F is a *-homomorphism is almost identical to the proofof proposition 4.2. Norm-decreasing follows by a simple calculation. Thatthe image lies in C0(G) is slightly less trivial, and takes some developmentof Gelfand theory of nonunital Banach algebras, for a proof that this is truesee Folland [1, p.15]. The last part, i.e. the fact that the image of F is densein C0(G) follows by the Stone Weierstrass theorem. �

If we recall the definition of the group algebra from section 3, i.e. we found acopy of L1(G) inside M(G), one might wonder if there is a generalization ofthe Fourier transform to the whole space M(G), indeed there is. It is calledthe Fourier-Stieltjes transform and is defined as follows. Let µ ∈M(G), thenthe Fourier-Stieltjes transform is the bounded continuous function µ on Gdefined by

µ(ξ) =

∫G

(x, ξ) dµ(x).

We check if it respects convolution:

µ ∗ ν(ξ) =

∫G

(xy, ξ) dµ(x)dν(y)

=

∫G

(x, ξ) (y, ξ) dµ(x)dν(y) = µ(ξ)ν(ξ)

It turns out that the Fourier-Stieltjes transform on M(G) is more interestingin our setting and we will see it in Bochner’s theorem in section 5.3. Wedefine the Fourier-Stieltjes transform on M(G) as

φµ(x) =

∫G

(x, ξ) dµ(ξ) (5.2)

for µ ∈M(G) and x ∈ G.

Proposition 5.4. The Fourier-Stieltjes transform on M(G) is a norm-decreasing linear injective map to the space of bounded continuous functionson G with the uniform norm.

Proof. All but the injectivity is trivial. Suppose φµ = 0 and f ∈ L1(G) then

0 =

∫Gf(x)φµ(x) dx =

∫G

∫Gf(x)(x, ξ) dµ(ξ)dx =

∫Gf(ξ−1) dµ(ξ).

But then µ = 0 since the image of F is dense in C0(G). �

5.2 FUNCTIONS OF POSITIVE TYPE

Let G be a locally compact abelian group. We define φ ∈ L∞(G) to be afunction of positive type if∫

G(f∗ ∗ f)φdx ≥ 0 ∀f ∈ L1(G).

Note that we can rewrite this as∫G

(f∗ ∗ f)φdx =

∫G

∫Gf∗(x)f(y−1x)φ(x) dydx

=

∫G

∫Gf(y−1)∆(y−1)f(y−1x)φ(x) dydx =

∫G

∫Gf(y)f(yx)φ(x) dydx


and finally

=

∫G

∫Gf(y)f(x)φ(y−1x) dydx ≥ 0 ∀f ∈ L1(G) (5.3)

which will be a useful form for the proofs. Define

P(G) = {φ ∈ L∞(G) : φ is of positive type} ∩ C(G)

Proposition 5.5. If φ ∈ P(G) then φ ∈ P(G).

Proof. This essentially follows from the fact that the complex conjugate ofan integral is the integral of the complex conjugate.∫

G(f∗ ∗ f)φdx =

∫G

(f∗ ∗ f)φdx ≥ 0 ∀f ∈ L1(G)

and we’re done. �

Proposition 5.6. If ρ : G → Hρ is a unitary representation and u ∈ Hρ,and if φ(x) = 〈ρ(x)u, u〉 then φ ∈ P(G).

Proof. Clearly, continuity of ρ implies continuity of φ, and

φ(y−1x) = 〈ρ(x)u, ρ(y)u〉

thus for f ∈ L1(G) we find that (5.3) becomes

0 ≤∫G

∫Gf(y)f(x)φ(y−1x) dydx

=

∫G

∫G〈f(x)ρ(x)u, f(y)ρ(y)u〉 dydx = ‖ρ(f)u‖2Hρ .

�

For the next result recall that f∗(x) = ∆(x−1)f(x−1) denotes involution of f ,but here G is unimodular, i.e. ∆ ≡ 1. Furthermore L2(G) ⊆ L1(G) followsby

‖f‖L2 ≤ ‖f‖L1 (∀f ∈ L1(G))

thus involution is well-defined for L2(G) functions.

Corollary 5.6. If f ∈ L2(G) then f ∗ f∗ ∈ P(G).

Proof. Let ρ(x) = Lx be the left regular representation, then

〈Lxf, f〉 =

∫Gf(x−1y)f(y) dy =

∫Gf(x−1y)f(y) dy = f ∗ f(x−1)

hence f ∗ f∗ ∈ P(G) by propositions 5.5 and 5.6. �

13 TOBIAS MATTSSON

5.3 BOCHNER’S THEOREM AND THE FIRST FOURIER INVERSIONTHEOREM

Bochner’s theorem will give us the necessary correspondence between positivecomplex Radon measures and functions of positive type, so that we can provethe Fourier inversion theorem and the Plancherel theorem. The general caseis due to Andre Weil, but was first proved by Gustav Herglotz for T and bySalomon Bochner for R, whom was the first to establish a place for functionsof positive type in harmonic analysis.

Theorem 5.7 (Bochner’s Theorem). Let G be a LCA Group. A functionφ on G is of positive type if and only if there is a (unique) non-negativemeasure µ ∈M(G) such that φ(x) = φµ(x)

Proof. (⇐= ) Assume µ ∈ M(G) is a non-negative measure and f ∈ L1(G)then∫

G

∫Gf(y)f(x)φ(y−1x) dydx =

∫G

∫G

∫Gf(y)f(x)(y−1x, ξ) dµ(ξ)dydx

=

∫G

∫G

∫Gf(y)(y, ξ)f(x)(x, ξ) dµ(ξ)dydx

=

∫G|f(ξ−1)|2 dµ(ξ) ≥ 0

i.e. the condition (5.3).( =⇒ ) We will manufacture a continuous linear functional on C0(G) andapply the Riesz-Markov theorem.

Assume φ ∈ P(G). Since ‖φ‖∞ = φ(1) (see Folland [1, p.79]) for all functionsof positive type it suffices to assume that φ(1) = 1. Now, if φ 6= 0 is a positivefunction then it induces a positive Hermitian form

〈f, g〉φ =

∫Gg∗ ∗ fφ dx

on L1(G) which clearly satisfies

|〈f, g〉φ| ≤ ‖φ‖∞‖f‖1‖g‖1

by applying the Schwarz-inequality. With our assumption on φ we get|〈f, g〉φ|2 ≤ 〈f, f〉φ〈g, g〉φ.

Now let g be an approximate identity {eλ}. Then (by definition) limλ∈Λ‖eλ ∗f − f‖1 = 0, hence |〈f, eλ〉φ| →

∫G φf dx. If the support of eλ is in U then

the support of e∗λ ∗ eλ is in U−1U , and 〈eλ, eλ〉φ = |∫G eλ dx|

2 = 1, hencee∗λ ∗ eλ is an approximate identity and 〈eλ, eλ〉φ dx → ‖φ‖∞ = φ(1) which is1 by our assumption.

Applying the Schwarz-inequality and the previous discussion to 〈f, eλ〉φ andtaking limits gives ∣∣∣ ∫

Gφf dx

∣∣∣2 ≤ 〈f, f〉φ. (5.4)


Now define h∗n = h ∗ · · · ∗ h n times, and let h = f∗ ∗ f then clearly h∗ = h.Apply (5.4) to the sequence f, h, h∗2, . . . , then we get∣∣∣ ∫

Gφh∗2

n−1dx∣∣∣ 12n ≤

∣∣∣ ∫Gφh∗2

ndx∣∣∣ 12n+1 ≤ ‖h∗2n‖

12n+1

1

and by the spectral radius formula and the fact that the range of the Fouriertransform is the spectrum of the function, it follows that

limn→∞

‖h∗2n‖1

2n+1

1 = ‖h‖12∞ = ‖|f |2‖

12∞ = ‖f‖∞.

Thus f →∫G φf induces a linear functional A : f →

∫G φf on F(L1(G)).

And since F(L1(G)) is dense in C0(G) by the Stone-Weierstrass theorem, itextends to a linear functional there, of norm ≤ 1. Thus by the Riesz-Markovrepresentation theorem there is a measure µ ∈M(G) such that

A(f) =

∫Gf(ξ) dµ(ξ) =

∫G

∫Gf(x)(x, ξ) dµ(ξ)dx

but then φ(x) =∫G

(x, ξ) dµ(ξ−1). Thus φ(1) = µ(G) ≤ ‖µ‖ ≤ 1, it followsthat µ ≥ 0. �

Now define the Bochner space B(G) = {φµ : µ ∈M(G)}. Bochner’s theoremsays that B(G) = spanCP(G). We also define µφ ∈ M(G) to correspond toφ under (5.2), i.e. it is the inverse map φ 7−→ µφ of (5.2).

Before we prove the Fourier inversion theorem we need a lemma

Lemma 5.8. If U ⊆ G is compact then there exists a compactly supportedfunction f ∈ P(G) such that f ≥ 0 on G and f > 0 on U .

Proof. Pick h ∈ Cc(G) such that∫G h = 1. Then h∗ ∗ h = |h|2, h∗ ∗ h ≥ 0

and h∗ ∗ h(1G

) = 1, thus there is a neighborhood U ′ of 1G

in G on which

h∗ ∗ h > 0.

By compactness we can cover U with finitely many of these. Let the trans-lates be given by ξ1, . . . ξk, and define f(x) =

∑i≤k ξih

∗ ∗ h(x), thus f(ξ) =∑i≤k Lξi h∗ ∗ h(ξ), which satisfies the conclusion. The convolution of two

L2(G) functions is continuous, thus f is, and since∫Gψ∗ ∗ ψf(x) dx =

∑i≤k

∫G

(ξiψ)∗ ∗ (ξiψ)h∗ ∗ h(x) dx ≥ 0

for all ψ ∈ L1(G), we have f ∈ P(G). �

Theorem 5.9 (First Fourier Inversion Theorem). Suppose dual Haar mea-sure dξ is suitably normalized relative to dx. If f ∈ B(G) ∩ L1(G) thenf ∈ L1(G) and

f(x) =

∫G

(x, ξ)f(ξ) dξ. (5.5)

15 TOBIAS MATTSSON

Proof. Again, we will construct a positive linear functional on Cc(G) wherewe can apply the Riesz representation theorem.

So let ψ ∈ Cc(G), then with U = supp(ψ) we apply lemma 5.8, and get somef ∈ Cc(G) ∩ P(G) ⊆ L1(G) ∩ P(G) with f > 0 on U . If g is another suchfunction (as f) and h ∈ L1(G) then∫

Gh dµg =

∫G

∫Gh(x)(x, ξ) dxdµg(ξ) =

∫Gh(x)f(x−1) dx = h ∗ f(1)

substituting h for h ∗ g or h ∗ f we get∫G hg dµf =

∫G hf dµg, now recalling

that F(L1(G)) is dense in C0(G) we ge that f dµg = g dµf .

Now define I : Cc(G) → C, ψ 7−→∫Gψ

fdµf , clearly I is a positive linear

functional, by the above discussion we get∫G

ψ

fdµf =

∫G

ψ

f gg dµf =

∫G

ψ

f gf dµg =

∫G

ψ

gdµg

therefore I(ψ) is invariant of the choice of f . And by the same discussion weget that I(gψ) =

∫G ψdµg, hence for appropriate ψ, f , I 6≡ 0.

Now we show translation invariance. Let η ∈ G be a character, then∫G

(x, ξ) dµf (ηξ) = (x, η)f(x) = (ηf)(x)

hence dµf (ηξ) = dµηf (ξ). Hence by translation of F we have for f such that

f > 0 on U ∩ supp(Lη(ψ)) that

I(Lηψ) =

∫G

ψ(η−1ξ)

fdµf (ξ) =

∫G

ψ(ξ)

f(ηξ)dµf (ηξ) =

∫G

ψ(ξ)

ηf(ξ)dµηf (ξ) = I(ψ).

Next, we note that it also follows that I(ψ) =∫G ψ(ξ) dξ which is the regular

dual Haar measure. And by our previous choice-independence of f we havethat I(ψf) =

∫G ψ dµf , hence f(ξ) dξ = dµf (ξ), thus f ∈ L1(G) and∫

G(x, ξ)f(ξ) dξ =

∫G

(x, ξ)dµf (ξ) = f(x) (5.6)

by Bochner’s theorem, and we’re done. �

The Fourier Inversion theorem is essentially one-half of a duality statementabout ”sufficiently nice” functions f : G→ C determining precisely another”sufficiently nice” function f : G → C. If one has ”very-sufficiently-nice”functions this relationship becomes exact and they determine one-anothercompletely, we make this more precise in the next section.

5.4 PLANCHEREL THEOREM

The Plancherel theorem says that the L2(G) and L2(G) are indistinguishableas Hilbert spaces, and later any decomposition of either gives a decompositionof the other. This is the fundamental L2 duality theorem in other words andjust one of many dualities in Harmonic Analysis.


Theorem 5.10 (Plancherel Theorem, 1910). The Fourier transform is ex-tendable to a unitary isomorphism between L2(G) and L2(G), in particularit is an isometry:

‖f‖2L2(G) =

∫G|f(x)|2 dx =

∫G|f(x)|2 dξ = ‖f‖2

L2(G).

Proof. Let f ∈ L1(G) ∩ L2(G) then f ∗ f∗ ∈ L1(G) ∩ P(G) by the same

argument as in lemma 5.8, and f ∗ f∗ = |f |2, so by the inversion theorem∫G|f |2 dx = f ∗ f(1) =

∫Gf ∗ f∗(ξ) dξ =

∫G|f |2 dξ

hence the Fourier transform is an isometry in the L2-norm and extendsuniquely to an isometry L2(G) → L2(G). We already have injectivity soit suffices to show surjectivity. Let ψ ∈ L2(G) be orthogonal in the L2-inner-product against all f with f as before, then

〈ψ, Lxf〉 = 0 =

∫G

(x, ξ)ψ(ξ)f(ξ) dξ.

But ψf ∈ L1(G), since ψ, f ∈ L2(G), hence ψ(ξ)f(ξ) dξ ∈ M(G). It the

follows by injection of F that ψf = 0 a.e., but lemma 5.8 says f > 0 henceψ = 0 a.e. And we’re done. �

We have now established the Fourier transform on L1(G) +L2(G) hence it isnatural to ask what happens for general Lp-spaces. In the case of 1 ≤ p ≤ 2we have Lp(G) ⊆ L1(G) + L2(G) which means that we can define f for f ∈Lp(G). Moreover, with the help of the Riesz-Thorin interpolation theoremone can prove the Hausdorff-Young inequality (see Folland [1, p.100]). Inthe case of p > 2, it becomes technical as one needs help from the theory ofdistributions, this exceeds the scope of this paper but is recommended forthe curious reader as a continuance of this section.

6 PONTRYAGIN DUALITY

Pontryagin Duality is a special case of Stone-Gelfand duality, nevertheless itgives a dual relation on the category of locally compact abelian groups calledthe double dual functor, this functor is covariant and naturally isomorphicto the identity functor, i.e. given a locally compact group G the charactergroup G = hom(G,T) is again locally compact given the topology of compactconvergence, and Pontryagin duality states that

G ∼= G.

There are further subcategory dualities which we will present soon, but fornow let us prove the main result.

Consider the map Φ : G → G defined by (ξ,Φ(x)) = (x, ξ) for all x ∈ G.This map Φ turns out to have some strong qualities, both topological andgroup theoretic.

Let B(G) ∩ L1(G) = B1(G).

17 TOBIAS MATTSSON

Lemma 6.1. The linear span of P(G) ∩Cc(G) contains all functions of theform f ∗ g for f, g ∈ Cc(G), moreover it is dense in Cc(G) in the uniformnorm and dense in Lp(G) in the Lp-norm for any 1 ≤ p <∞.

Proof. By corollary 5.6, P(G) ∩ Cc(G) includes all functions of the formf ∗ f(x−1) for f ∈ Cc(G). By the polarization identity we get that

f ∗ h(x−1) =1

4[(f + h) ∗ (f + h)(x−1)− (f − h) ∗ (f − h)(x−1)

+ i((f + h) ∗ (f + h)(x−1)− (f − ih) ∗ (f − ih)(x−1))]

is in P(G)∩Cc(G) for all f, h ∈ Cc(G), hence functions of the form f ∗h arein P(G)∩Cc(G). Now, {f ∗g : f, g ∈ Cc(G)} is dense in Cc(G) in the uniformnorm and dense in Lp(G) in the Lp-norm, thus concluding the proof. �

From this lemma it clearly follows that B(G) contains all functions of theform f ∗ g for f, g ∈ Cc(G), moreover B1(G) is dense in Cc(G) in the uniformnorm and dense in L1(G) in the L1-norm.

Lemma 6.2. If φ, ψ ∈ Cc(G) then φ ∗ ψ = f for some f ∈ B1(G). AndF(B1(G)) is dense in Lp(G) for all p <∞.

Proof. Let f(x) = φ(x) and g(x) = ψ(x) and h(x) = φ ∗ ψ(x), then clearly,

h(x) =

∫ ∫(x, ξ)φ(ξη−1)ψ(η) dξdη

=

∫ ∫(x, ξη)φ(ξ)ψ(η) dξdη

=

∫ ∫(x, ξ)(x, η)φ(ξ)ψ(η) dξdη = f(x)g(x)

Now f, g, h ∈ B(G) since φ, ψ, φ ∗ ψ ∈ L1(G). If k ∈ L1(G) ∩ L2(G), then itfollows by the Plancherel theorem 5.10 that∣∣∣∣∣∫Gfk

∣∣∣∣∣ =

∣∣∣∣∣∫G

∫G

(x, ξ)φ(x)k(x) dξdx

∣∣∣∣∣ =

∣∣∣∣∣∫Gφk

∣∣∣∣∣ ≤ ‖φ‖2‖k‖2 = ‖φ‖2‖k‖2.

Thus f ∈ L2(G), and by the same argument one notes that g ∈ L2(G). Thisimplies that h = fg ∈ L1(G), in particular h ∈ B1(G). Now the Fourierinversion theorem 5.9 gives

h(x) =

∫G

(x, ξ)h(ξ) dξ

thus injectivity of the Fourier transform gives h = φ ∗ψ and we’re done. �

We’re now ready to prove Pontryagin’s theorem.

Theorem 6.3 (The Pontryagin Duality Theorem). The map Φ : G → G is

a group isomorphism and a homeomorphism.

The general case is due to Egbert van Kampen, while the compact and secondcountable case was proved by Lev Pontryagin.


Proof. Let x, y ∈ G and ξ ∈ G, then

(ξ,Φ(xy)) = (xy, ξ) = (x, ξ)(y, ξ) = (ξ,Φ(x))(ξ,Φ(y)) = (ξ,Φ(x)Φ(y))

thus Φ(xy) = Φ(x)Φ(y). Injectivity follows by the Gelfand-Raikov theorem

(see Folland [1, p.84]) immediately. Thus Φ is an isomorphism intoG. There

are three more things to prove:

1. Φ is a homeomorphism.

2. Φ(G) is closed inG.

3. Φ(G) is dense inG.

Next let x ∈ G and {xα}α∈A be a net in G, then clearly xα → x implies thatf(xα)→ f(x) for all f ∈ B1(G), on the other hand if xα 6→ x then there is aneighborhood U of x and a set B ⊆ A, such that for every α ∈ A there is aβ ∈ B such that α ≤ β,4 for which xβ 6∈ U for β ∈ B. By lemma 6.1 there isa f ∈ B1(G) such that supp(f) ⊂ U and f(x) 6= 0, hence f(xα) 6→ f(x).

Now by the Fourier inversion theorem 5.9 we get that f(xα) → f(x) isequivalent to ∫

G(xα, ξ)f(ξ) dξ →

∫G

(x, ξ)f(ξ) dξ (6.1)

thus this is equivalent to xα → x by our previous discussion. But equation(6.1) clearly says that∫

GΦ(xα)f(ξ) dξ →

∫G

Φ(x)f(ξ) dξ (∀f ∈ F(B1(G))). (6.2)

Now ‖Φ(xα)‖∞ = 1 for all α ∈ A since the topology ofG is the weak-*

topology of L∞(G). Moreover lemma 6.2 says that F(B1(G)) is dense inL1(G), thus it follows that equation (6.1) is equivalent to equation (6.2) andby transitivity of equivalence equation (6.2) is equivalent to xα → x. Inconclusion, Φ is a homeomorphism.

By (1) Φ(G) is locally compact in the relative topology. Let ξ0 ∈ Φ(G), andlet U be a neighborhood of ξ0 whose closure is compact. Then Φ(G) ∩ Uis compact and hence closed in

G, but ξ0 ∈ Φ(G) ∩ U and it follows that

ξ0 ∈ Φ(G). Thus Φ(G) is closed.

Suppose Φ(G) isn’t dense inG, then there is a function F ∈ F(G) which is

0 at every point of Φ(G) but not identically 0 (see theorem 1.6.4 in Rudin[2, p.27]). So there exists φ ∈ L1(G) such that

F (ξ) =

∫Gφ(ξ)(ξ, ξ) dµ(ξ).

Since F (α(x)) = 0 for all x ∈ G we get∫Gφ(ξ)(ξ,Φ(x)) dµ(ξ) =

∫Gφ(ξ)(x, ξ) dµ(ξ) = 0

from which one sees that φ = 0 by proposition 5.4. This implies that F = 0,and this contradiction completes the proof. �

4A set B with such a property is called cofinal

19 TOBIAS MATTSSON

The following results are essentially corollaries of Pontryagin duality, buttheir importance render them theorems. They can be seen as completing thepicture that we have been drawing so far.

Theorem 6.4 (Second Fourier Inversion Theorem). If f ∈ L1(G) and f ∈L1(G), then

f(x) =

∫G

(x, ξ)f(ξ) dξ for a.e. x ∈ G.

Proof. We have f ∈ B(G) ∩ L1(G) because

f(ξ) =

∫G

(x, ξ)f(x) dx =

∫G

(x, ξ)f(x−1) dx =

∫G

(x, ξ) dµf(x).

Thus by the first Fourier inversion theorem

f(x−1) =

∫G

(x−1, ξ)f(ξ) dξ for a.e. x ∈ G.

�

Theorem 6.5 (Uniqueness theorem). If µ, ν ∈ M(G) such that µ = ν thenµ = ν.

Proof. By proposition 5.4 and by Pontryagin duality we get that µ is com-pletely determined by φµ(ξ) = µ(ξ−1). �

A nice fact about Pontryagin duality is that it gives a dual relationshipbetween two topological notions,this is the subcategory duality we mentionedabove.

Proposition 6.1. The dual G of G is discrete if and only if G is compact.And G is discrete if and only if the dual group G is compact.

Proof. Assume G is discrete, then L1(G) has a unit δ1 (the point mass mea-sure at x = 1). Hence its spectrum G is compact. Now assume G is compactthen the constant function 1 is in L1(G) so

{f ∈ L∞(G) :∣∣∣ ∫

Gf dx

∣∣∣ > 1

2}

is a weak* open set. Now, it follows by proposition 5.2 that for every ξ ∈ Gwe have that∫

ξ = 1 (if ξ = 1),

∫ξ = 0 (if ξ = 0).

Thus {1} is an open set in G, so G is discrete. The other directions followsby Pontryagin duality. �


7 THE CULMINATION: DECOMPOSITION OF UNITARYREPRESENTATIONS

In this section we will describe a way of decomposing unitary representationsof locally compact first countable abelian groups, but first we will need tocover some theory on projection-valued measures.

Projection valued-measures have broader applications in functional analysis,in particular in a generalized version of the spectral theorem to infinite di-mensional spaces, there is more on this topic in Folland [1, p.16-26]. Let H bea Hilbert space and let X be a topological group with Borel σ-algebra B(X),furthermore let P(H) denote the orthogonal projections onto subspaces of H.Then a map P : B(X)→ P(H) is called a H-projection valued measure if itsatisfies:

1. (P1) P (X) = I

2. (P2) If {Ei}i∈A ⊂ B(X) are pairwise disjoint then

P (⋃i∈A

Ei) =∑i∈A

P (Ei) (A countable)

with convergence in the strong operator topology.

The interesting thing about these maps P is that they give rise to familiesof complex measures µu,v, we prove this fact.

Proposition 7.1. Let P : B(X)→ P(H) be a H-projection valued measure,then µu,v(E) = 〈P (E)u, v〉 is a complex measure for all u, v ∈ H.

Proof. Let u, v ∈ H. Nonnegativity is clearly induced by the inner-product.Countable additivity follows from the calculation:

µu,v(⋃i∈A

Ei) = 〈P (⋃i∈A

Ei)u, v〉 = 〈∑i∈A

P (Ei)u, v〉

=∑i∈A〈P (Ei)u, v〉 =

∑i∈A

µu,v(Ei).

And since P (X) = I implies that P (X) = P (X ∪ ∅ ∪ · · · ) = I + P (∅) + · · ·we get P (∅) = 0 (projection onto {0}) and µu,v(∅) = 0. �

A projection valued measure P is called regular if the corresponding complexmeasure is regular. The projection valued measures share many propertieswith normal complex measures, one particularly interesting property is thatof having a well defined notion of integration, i.e. one can integrate functionswith projection valued measures, we shall define this more precisely now. Letf be a bounded and measurable function on (X,B(X)), then we have for allv ∈ H ∣∣∣∣∣

∫Xf dµv,v

∣∣∣∣∣ ≤ ‖f‖∞‖µv,v‖ = ‖f‖∞µv,v(X) = ‖f‖∞‖v‖2H

21 TOBIAS MATTSSON

thus by polarization, if u, v ∈ H such that ‖u‖ = ‖v‖ = 1, then∣∣∣∣∣∫Xf dµv,v

∣∣∣∣∣ ≤ 4‖f‖∞.

It follows that ∣∣∣∣∣∫Xf dµv,v

∣∣∣∣∣ ≤ 4‖f‖∞‖u‖‖v‖.

Thus the Frechet-Riesz representation theorem gives that there is some boundedoperator T on H such that

〈Tu, v〉 =

∫Xf dµu,v.

We symbolically represent T by∫X f dP . This is how we define integration

of H-projection valued measures. One could also define them by approxi-mating f by simple functions since every Borel measurable function f canbe approximated as a uniform limit by simple functions, just as one does inordinary integration theory.

Before we can prove the main theorem of this section we need to definethe Gelfand transform and prove the following theorem on projection-valuedmeasures.

Let x ∈ G and let A be a Banach algebra. Define the function x : σ(A)→ Cby x(f) = f(x) for every f ∈ σ(A). Then the map x → x is called theGelfand map.

Theorem 7.1. If A is a commutative nonunital Banach *-algebra and φ anondegenerate *-representation of A on the Hilbert space H, then there is aunique regular projection-valued measure P on the spectrum σ(A) of A suchthat

φ(x) =

∫Xx dP (∀x ∈ A)

where x is the Gelfand transform.

Proof. Let B be the norm-closure of φ(A) in B(H), this is a C∗ subalgebra ofB(H) since φ is a *-representation. There are two cases we need to consider,either I ∈ B (the identity operator) or not, we consider the former first. Now,φ induces a map φ∗ : σ(B)→ σ(A) called the pullback of φ, which is definedas φ∗h = h(φ) for all h ∈ B. If φ∗H1 = φ ∗ h2 then h1, h2 agree on σ(A) andtherefore everywhere, thus φ∗ is injective. Since σ(B) is a compact Hausdorffspace, φ∗ is a homeomorphism onto its range, which then is a compact subsetof σ(A). And then the Spectral theorem (see Folland [1, p.22 (top)] for areference) associates a unique regular projection valued measure P0 to B onσ(B) such that

T =

∫T dP0 (∀T ∈ B).

The pullback of P0 by φ∗ induces a projection valued measure on σ(A), moreprecisely P (E) = P0(φ∗−1(E)) (it is an easy check to see that P is regular,


it follows by the properties of φ). And finally the Gelfand transforms on Aand B are related by φ(x)(h) = x(φ∗h), and therefore

φ(x) =

∫x(φ∗h) dP0(h) =

∫x dP (∀x ∈ A).

For the second case when I 6∈ B, the proof is next to identical, with the onlydifference being that one considers the unital augmentation A = A×C whereφ(x, z) = φ(x) + zI and B = B ⊕ CI. For a more detailed discussion on thissee Folland [1, p.27 (bottom)]. �

Theorem 7.2. Let H be some Hilbert space, G a locally compact first count-able abelian group, moreover let ρ : G → U(Hρ) be a unitary representation

G, then there is a unique regular Hρ-projection-valued measure P on G suchthat

ρ(x) =

∫G

(x, ξ) dP (ξ) (x ∈ G)

ρ(f) =

∫Gξ(f) dP (ξ) (f ∈ L1(G))

Proof. Since G can be identified with the spectrum of L1(G) we get by the-orem 7.1 (with A = L1(G)) that there is a unique Hρ-projection valuedmeasure such that

ρ(f) =

∫Gξ(f) dP (ξ) (f ∈ L1(G)).

By the proof of theorem 4.6 ρ(x) is the strong limit of ρ(Lxψα) for some ap-proximate identity {ψα}α∈A (taken as a sequence since G is first countable),thus

ρ(Lxψα) =

∫Gξ(Lxψα) dP (ξ) =

∫G

(x, ξ)ξ(ψα) dP (ξ)

which converges to∫G

(x, ξ) dP (ξ) by the dominated convergence theorem

since |ξ(ψα)| ≤ 1 and ξ(ψα)→ 1 for every ξ ∈ G, thus we’re done.

�

As a finale to our discussion on abstract harmonic analysis we decomposethe left regular representation on L2(G). Let ρ(x)f(y) = f(x−1y), then

F−1[ρ(x)f ](ξ) =

∫G

(y, ξ)f(x−1y), dy =

∫G

(xy, ξ)f(y), dy = (x, ξ)F−1[f ](ξ).

Thus theorem 7.2 gives that F−1[P (E)f ](ξ) = χEF−1[f ], which implies that

P (E) = F(χEx)F−1.

Moreover,

ρ(f) =

∫Gξ(f)d[F(χEx)F−1].

23 TOBIAS MATTSSON

REFERENCES

[1] Gerald B Folland, A course in abstract harmonic analysis, vol. 29, CRCpress, 2016.

[2] Walter Rudin, Fourier analysis on groups, Courier Dover Publications,2017.

[3] Bruce E Sagan, The symmetric group: representations, combinatorial al-gorithms, and symmetric functions, vol. 203, Springer Science & BusinessMedia, 2013.

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