positivity in function algebras

IntroductionPositivity

Connections with Representation Theory

Positivity in Function Algebras

Jason Ekstrand

Intel Corporation

INFAS, March 2015

Jason Ekstrand Positivity in Function Algebras



What is a functional analyst doing at Intel?

Not functional analysis.

I I work on the Open-source 3-D graphics driver teamI Modern graphics cards are specialized processors that

perform moderate calculations millions of times persecond.

I My work has focused on the compiler for Intel GPUsI My work so far has been:

I 20% Graph TheoryI 15% Algebraic Identities/ReductionsI 65% Problem Solving and writing C Code




Overview

I IntroductionI Problem StatementI Notation

I PositivityI Positivity in the DiscI Positivity in the AnnulusI Positivity in more General Domains

I Connections with Representation TheoryI Future WorkI References




Problem StatementNotation

Problem Statement

Let A(D) be the disc algebra and give A(D) the involution

f 7→ f ∗; f ∗(z) = f (z)

This yields a Banach ∗-algebra that is not a C∗-algebra.

Properties of A(D, ∗)

I A(D) (without the involution) is a norm-closed subalgebraof C(T) so it is an operator algebra

I A(D, ∗) is a ∗-subalgebra of C[−1,1]

I For every f ∈ A(D), σ(f ) = f (D−)





We wish to study the positive elements of A(D, ∗).

Definition

Let A be a general ∗-algebra (no assumptions of norm). Thenthe set of positive elements of A, denoted A+, is given by

A+ =

{∑k

a∗kak : ak ∈ A

}.

Definition

Let A be a unital C∗-algebra. Then an element a ∈ A is said tobe positive if a∗ = a and σ(a) ⊆ R+.





What is a good definition of positivity in A(D, ∗)?

Definition

Let f ∈ A(D, ∗). Then f is said to be positive if

f ([−1,1]) ⊆ R+.

Is this the right definition?

Theorem (Ekstrand & Peters, 2013)

Let f ∈ A(D, ∗). Then f is positive (as defined above) if and onlyif f = g∗g for some g ∈ A(D).





Notation

For a domain G ⊆ C, we have the following algebras:I H(G) of holomorphic functions on GI H∞(G) of bounded holomorphic functions on GI A(G) of bounded holomorphic functions on G which have

continuous extension to G−

If f : G→ C and rT ⊆ G and, we define the function

fr : [−π, π]→ C; fr (t) = f (reit ).

When it makes sense, we define the pth Hardy space

Hp(G) = {f ∈ H(G) : ‖fr‖p is bounded in r}




Positivity in the DiscPositivity in the AnnulusGeneralizations

Positivity in the Disc

We begin with the case of non-vanishing functions.

Let f ∈ A(D) be non-vanishing. Since D is simply connected,

f (z) = eh(z) for some h ∈ H(D).

However, h need be neither bounded nor continuous on D−.

Lemma

Suppose h : D→ C is continuous and that there is a continuousfunction F : D− → C with F = eh on D. If K is the set of zerosof F on T then h can be continuously extended to D− \ K .





Theorem

Let f ∈ H(D) be positive with no roots in D. Then, for everyinteger n > 0 there is a unique positive function g ∈ H(D) suchthat f = gn. If f ∈ Hp(D) for some 1 ≤ p ≤ ∞, then g ∈ Hnp(D).If f ∈ A(D), then g ∈ A(D).

Sketch of proof.

I f = eh for some h ∈ H(D); let g = eh/n on DI Define x : T→ C as x = eh/n on T \ K and x = 0 on KI Then x is continuous on T and x is a.e. the boundary

values of g so g ∈ A(D).





BSF Factorization

For any function f ∈ H1(D), we can write f = BSF where

F (z) = λexp[

12π

∫ π

−π

eiθ + zeiθ − z

log |f (eiθ)| dθ],

for some λ ∈ C with |λ| = 1 and

B(z) = zp0

∞∏n=1

[αn

|αn|αn − z1− αnz

]pn

where {αn} are the roots of f with multiplicities pn and

S(z) = exp[−∫

eiθ + zeiθ − z

dµ(θ)

]for some singular positive measure µ on [−π, π].





If we are going to use the BSF factorization, we need to handlethe positivity and continuity of the different pieces.

Theorem

Let f ∈ A(D) and decompose f as f = gB where g ∈ H∞(D)and B is a Blaschke product. Then g ∈ A(D) and g has thesame zeros on T as f .

Theorem

Let f ∈ A(D) and let B be a Blaschke product such that f (z) = 0whenever z is a limit point of the roots of B. Then fB ∈ A(D).





Theorem

Let B be the Blaschke product. If B has the same roots assome positive f ∈ H(D), then there is another Blaschke productB+ with B = B∗+B+.

Theorem

Let f ∈ Hp(D) for some 1 ≤ p ≤ ∞. Then f is positive if andonly if there exists g ∈ H2p(D) so that f = g∗g. If f ∈ A(D) theng may also be chosen to be in A(D).





Positivity in the Annulus

Definition

Fix 0 < r0 < 1 and define the annulus

A = {z ∈ C : r0 < |z| < 1}.

We define the following algebras:I H(A) of all holomorphic functions on A,I Hp(A) of all holomorphic functions on A with ‖fr‖p bounded

for r0 < r < 1,I A(A) of all holomorphic functions on A with continuous

extension to A−.





Properties of H(A)

Given a function f ∈ H(A), we have the Laurent series

f (z) =∞∑

n=−∞anzn =

∞∑n=0

anzn +∞∑

n=1

a−n1zn

so f (z) = g(z) + h(r0/z) where g,h ∈ H(D).

Observation

I f ∈ Hp(A) if and only if g,h ∈ Hp(D)

I f ∈ A(A) if and only if g,h ∈ A(D)

I f ∈ Hp(A) can be recovered from its boundary values





Positivity in HP(A)

Definition

Let f ∈ H(A). Then f is said to be positive if

f (x) ≥ 0 for all x ∈ A ∩ R.

I How do we study positive functions on A?I For f ∈ H(A), f (z) = g(z) + h(r0/z) where g,h ∈ H(D).

However, f positive does not imply that g or h is positive.I f ∈ H(A) non-vanishing does not imply f = eg .I How do we replace our use of the BSF factorization?





Non-vanishing functions in H(A)

The problem here is that A is not simply connected.

Theorem

Let G be a domain and f be holomorphic on G. Suppose f isnon-vanishing and ∮

γ

f ′(z)

f (z)dz = 0

for every simple closed curve γ. Then there exists aholomorphic function g on G so that f = eg .





Definition

For f ∈ H(A) non-vanishing, define the winding number of f by

wn(f ) =1

2πi

∮γr

f ′(z)

f (z)dz

where γr (t) = reit for t ∈ [−π, π] and r0 < r < 1.

Theorem (Ekstrand, 2014)

Let f ∈ H(A) be positive and non-vanishing. Then wn(f ) is aneven number.

For any positive f ∈ H(A), the function g(z) = f (z)z−wn(f ) ispositive with wn(g) = 0.






Let f ∈ H(A) be positive and non-vanishing. Then there existsa function g ∈ H(A) so that f = g∗g. Furthermore, if f ∈ Hp(A),then g ∈ H2p(A) for 1 ≤ p ≤ ∞ and, if f ∈ A(A), then g ∈ A(A).

Sketch of proof.

I Let f0(z) = f (z)z−wn(f ); wn(f0) = 0.I f0 = eh for some h ∈ H(A).I Define g by g(z) = eh(z)/2zwn(f )/2.I Continuity is similar to the disc case.





Hp spaces of an annulus (Sarason, 1965)

In his 1965 work, Sarason studies holomorphic functions on Aand tries to recover a BSF factorization for the annulus.

I Sarason’s work focuses on the universal covering surface

A = {(r , t) ∈ R2 : r0 < r < 1}

with the covering map

ϕ : A→ A; ϕ(r , t) = reit .

I Sarason develops a BSF factorization for modulusautomorphic functions A

I Unfortunately, these result don’t translate easily to Hp(A)





Blaschke Products on A

Sarason’s construction is enough to get us the following:

Theorem (Sarason, 1965; Ekstrand, 2014)

Let f ∈ H∞(A) that is not identically zero and let {an}∞n=1 be theset of zeros of f repeated according to multiplicity. Then

∞∑n=1

min(

1− |an|,1−r0

|an|

)<∞.






Let f ∈ H∞(A) that is not identically zero and let {an} be the roots of frepeated according to multiplicity. Then the Blaschke products

B1(z) =∏

|an|≥√

r0

an

|an|an − z

1− anzand B2(z) =

∏|an|<

√r0

an

|an|r0/an − z

1− (r0/an)z

converge and we may decompose f as f (z) = g(z)B1(z)B2(r0/z)where g is bounded, holomorphic, and non-vanishing on A. If f has acontinuous extension to A− then so does g.


An element f ∈ Hp(A) is positive if and only if f = g∗g for someg ∈ H2p(A). Furthermore, if f is continuous on A−, then g maybe chosen continuous on A−.





Generalizations to other domains

Definition

Let G be a domain. We say that G is symmetric if

G = G∗ = {z : z ∈ G}.


Let G be a symmetric domain where ∂G is the union of finitelymany disjoint Jordan curves and let f ∈ H∞(G). Then f ispositive if and only if there is some g ∈ H∞(G) so that f = g∗g.Furthermore, if f ∈ A(G) then g may be chosen in A(G).





Definition

Let A be a ∗-algebra. Then a ∗-representation of A is a pair(H, ϕ) where H is a Hilbert space and ϕ : A → B(H) is a∗-homomorphism.

What about A(G, ∗)?I If (H, ϕ) is a ∗-representation of A then, for all g ∈ A,ϕ(g∗g) = ϕ(g)∗ϕ(g) is positive in H.

I The one-dimensional ∗-representations of A(G, ∗) areexactly the point-evaluations on G ∩ R.

I f ∈ A(G, ∗) is positive if and only if ϕ(f ) ≥ 0 for everyone-diimensional ∗-representation ϕ of A(G, ∗).





Let G be a region so that ∂G is the union of finitely manydisjoint Jordan curves in C∞. For each f ∈ A(G), TFAE:

1. f is positive, i.e., f (G ∩ R) ≥ 0,2. f = g∗g for some g ∈ A,3. f =

∑ni=1 g∗i gi for some g1, . . . ,gn ∈ A,

4. f = limn→∞ fn where each fn is of the form given in 3.5. ϕ(f ) ≥ 0 for every one-dimensional ∗-rep. (C, ϕ) of A(G)

5. is equivalent to σ(a) ≥ 0 in abelian C∗-algebras




Future Work

1. Extend the results to even more general domainsI While the restriction that ∂G is the union of finitely many

disjoint Jordan curves is sufficient, I have no proof that it isnecessary.

I Unfortunately, such an extension would probably need anew technique.

2. Try and extend these results to a non-abelian caseI These definitions extend fairly easily toMn×n(A(G))

3. Consider domains not in C such as Riemann surfacesI There is a 1965 paper by Voichick and Zalcman that gives a

BSF factorization for a certain class of Riemann surfaces




References I

[1] Gert K. Pedersen, Analysis now, Graduate Texts in Mathematics,vol. 118, Springer-Verlag, New York, 1989. MR971256 (90f:46001)

[2] Gerard J. Murphy, C∗-algebras and operator theory, Academic Press,Inc., Boston, MA, 1990.

[3] Krzysztof Ciesielski, Set theory for the working mathematician, LondonMathematical Society Student Texts, vol. 39, Cambridge UniversityPress, Cambridge, 1997.

[4] John B. Conway, A course in functional analysis, 2nd ed., GraduateTexts in Mathematics, vol. 96, Springer-Verlag, New York, 1990.

[5] Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York,1973. McGraw-Hill Series in Higher Mathematics.

[6] Theodore W. Palmer, Banach algebras and the general theory of∗-algebras. Vol. 2, Encyclopedia of Mathematics and its Applications,vol. 79, Cambridge University Press, Cambridge, 2001. ∗-algebras.




References II

[7] John B. Conway, Functions of one complex variable, 2nd ed., GraduateTexts in Mathematics, vol. 11, Springer-Verlag, New York-Berlin, 1978.

[8] Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill BookCo., New York, 1987.

[9] Kenneth Hoffman, Banach spaces of analytic functions, Prentice-HallSeries in Modern Analysis, Prentice-Hall Inc., Englewood Cliffs, N. J.,1962.

[10] Donald Sarason, The Hp spaces of an annulus, Mem. Amer. Math. Soc.No. 56 (1965), 78.

[11] Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehrender Mathematischen Wissenschaften [Fundamental Principles ofMathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992.

[12] Rolf Nevanlinna, Analytic functions, Translated from the second Germanedition by Phillip Emig. Die Grundlehren der mathematischenWissenschaften, Band 162, Springer-Verlag, New York, 1970.




References III

[13] Michael Voichick and Lawrence Zalcman, Inner and outer functions onRiemann surfaces, Proc. Amer. Math. Soc. 16 (1965), 1200–1204.

[14] Stewart S. Cairns, An elementary proof of the Jordan-Schoenfliestheorem, Proc. Amer. Math. Soc. 2 (1951), 860–867.

[15] Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehrender Mathematischen Wissenschaften [Fundamental Principles ofMathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992.

[16] Ryuji Maehara, The Jordan curve theorem via the Brouwer fixed pointtheorem, Amer. Math. Monthly 91 (1984), no. 10, 641–643, DOI10.2307/2323369.

Thank You!




Idea Behind the Proof

Theorem (Riemann Mapping Theorem)

Let G ⊆ C be a simply connected region that is not the wholeplane and let a ∈ G. Then there is a unique holomorphicbijection φ : G→ D so that φ(a) = 0 and φ′(a) > 0.

Theorem (Caratheodory)

Let G ⊆ C be a simply connected region whose boundary is aJordan curve. Then the Riemann map φ : G→ D extends to ahomeomorphism Φ : G− → D−.




Sketch of Proof

Start with some symmetric region G and f ∈ H∞(G)

G




Sketch of Proof

Pick a single hole H in G

G

H




Sketch of Proof

Define a Caratheodory map φ : C \ H → D−

G

HG

φ φ(H)

φ(G)




Sketch of Proof

Pick r0 so that {z ∈ C : r0 ≤ |z| < 1} ⊆ φ(G)

H

G

G

φ φ(H)

φ(G)




Sketch of Proof

I This gives us an annulus A = {z ∈ C : r0 ≤ |z| < 1}.I We can factor f ◦ φ−1 as f ◦ φ−1 = gB where g ∈ H∞(A)

and B is a Blaschke product.I Translating back to G, f = (g ◦ φ)(B ◦ φ).I A similar trick can be used to ensure wn(f ◦ ϕ−1) = 0.I Decompose f , square root the non-vanishing part and put

it back together as we did before.I Thanks to the Caratheodory theorem, φ is a

homeomorphism of C \ H and D− so continuity followsfrom results in the annulus.


positivity in function algebras

Documents