Introduction
Positivity
Connections with Representation Theory
Positivity in Function Algebras
Jason Ekstrand
Intel Corporation
INFAS, March 2015
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
What is a functional analyst doing at Intel?
Not functional analysis.
I
I work on the Open-source 3-D graphics driver team
I
Modern graphics cards are specialized processors that
perform moderate calculations millions of times per
second.
I
My work has focused on the compiler for Intel GPUs
My work so far has been:
I
I
I
I
20% Graph Theory
15% Algebraic Identities/Reductions
65% Problem Solving and writing C Code
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Overview
I
Introduction
I
I
I
Problem Statement
Notation
Positivity
I
I
I
Positivity in the Disc
Positivity in the Annulus
Positivity in more General Domains
I
Connections with Representation Theory
I
Future Work
I
References
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Problem Statement
Notation
Problem Statement
Let A(D) be the disc algebra and give A(D) the involution
f 7→ f ∗ ;
f ∗ (z) = f (z̄)
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 subalgebra
of 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− )
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Problem Statement
Notation
We wish to study the positive elements of A(D, ∗).
Definition
Let A be a general ∗-algebra (no assumptions of norm). Then
the set of positive elements of A, denoted A+ , is given by
(
)
X
A+ =
ak∗ ak : ak ∈ A .
k
Definition
Let A be a unital C ∗ -algebra. Then an element a ∈ A is said to
be positive if a∗ = a and σ(a) ⊆ R+ .
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Problem Statement
Notation
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 only
if f = g ∗ g for some g ∈ A(D).
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Problem Statement
Notation
Notation
For a domain G ⊆ C, we have the following algebras:
I
H(G) of holomorphic functions on G
I
H ∞ (G) of bounded holomorphic functions on G
I
A(G) of bounded holomorphic functions on G which have
continuous extension to G−
If f : G → C and r T ⊆ G and, we define the function
fr : [−π, π] → C;
fr (t) = f (reit ).
When it makes sense, we define the pth Hardy space
H p (G) = {f ∈ H(G) : kfr kp is bounded in r }
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Positivity in the Disc
Positivity in the Annulus
Generalizations
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 continuous
function F : D− → C with F = eh on D. If K is the set of zeros
of F on T then h can be continuously extended to D− \ K .
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Positivity in the Disc
Positivity in the Annulus
Generalizations
Theorem
Let f ∈ H(D) be positive with no roots in D. Then, for every
integer n > 0 there is a unique positive function g ∈ H(D) such
that f = g n . If f ∈ H p (D) for some 1 ≤ p ≤ ∞, then g ∈ H np (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 D
I
Define x : T → C as x = eh/n on T \ K and x = 0 on K
I
Then x is continuous on T and x is a.e. the boundary
values of g so g ∈ A(D).
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Positivity in the Disc
Positivity in the Annulus
Generalizations
BSF Factorization
For any function f ∈ H 1 (D), we can write f = BSF where
Z π iθ
e +z
1
iθ
log |f (e )| dθ ,
F (z) = λ exp
2π −π eiθ − z
for some λ ∈ C with |λ| = 1 and
∞ Y
ᾱn αn − z pn
p0
B(z) = z
|αn | 1 − ᾱn z
n=1
where {αn } are the roots of f with multiplicities pn and
Z iθ
e +z
dµ(θ)
S(z) = exp −
eiθ − z
for some singular positive measure µ on [−π, π].
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Positivity in the Disc
Positivity in the Annulus
Generalizations
If we are going to use the BSF factorization, we need to handle
the 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 the
same zeros on T as f .
Theorem
Let f ∈ A(D) and let B be a Blaschke product such that f (z) = 0
whenever z is a limit point of the roots of B. Then fB ∈ A(D).
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Positivity in the Disc
Positivity in the Annulus
Generalizations
Theorem
Let B be the Blaschke product. If B has the same roots as
some positive f ∈ H(D), then there is another Blaschke product
∗B .
B+ with B = B+
+
Theorem
Let f ∈ H p (D) for some 1 ≤ p ≤ ∞. Then f is positive if and
only if there exists g ∈ H 2p (D) so that f = g ∗ g. If f ∈ A(D) then
g may also be chosen to be in A(D).
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Positivity in the Disc
Positivity in the Annulus
Generalizations
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
H p (A) of all holomorphic functions on A with kfr kp bounded
for r0 < r < 1,
I
A(A) of all holomorphic functions on A with continuous
extension to A− .
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Positivity in the Disc
Positivity in the Annulus
Generalizations
Properties of H(A)
Given a function f ∈ H(A), we have the Laurent series
f (z) =
∞
X
n
an z =
n=−∞
∞
X
n
an z +
n=0
∞
X
a−n
n=1
1
zn
so f (z) = g(z) + h(r0 /z) where g, h ∈ H(D).
Observation
I
f ∈ H p (A) if and only if g, h ∈ H p (D)
I
f ∈ A(A) if and only if g, h ∈ A(D)
I
f ∈ H p (A) can be recovered from its boundary values
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Positivity in the Disc
Positivity in the Annulus
Generalizations
Positivity in H P (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?
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Positivity in the Disc
Positivity in the Annulus
Generalizations
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 is
non-vanishing and
I 0
f (z)
dz = 0
γ f (z)
for every simple closed curve γ. Then there exists a
holomorphic function g on G so that f = eg .
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Positivity in the Disc
Positivity in the Annulus
Generalizations
Definition
For f ∈ H(A) non-vanishing, define the winding number of f by
I 0
1
f (z)
wn(f ) =
dz
2πi γr f (z)
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 an
even number.
For any positive f ∈ H(A), the function g(z) = f (z)z −wn(f ) is
positive with wn(g) = 0.
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Positivity in the Disc
Positivity in the Annulus
Generalizations
Theorem (Ekstrand, 2014)
Let f ∈ H(A) be positive and non-vanishing. Then there exists
a function g ∈ H(A) so that f = g ∗ g. Furthermore, if f ∈ H p (A),
then g ∈ H 2p (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)/2 z wn(f )/2 .
I
Continuity is similar to the disc case.
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Positivity in the Disc
Positivity in the Annulus
Generalizations
H p spaces of an annulus (Sarason, 1965)
In his 1965 work, Sarason studies holomorphic functions on A
and tries to recover a BSF factorization for the annulus.
I
Sarason’s work focuses on the universal covering surface
 = {(r , t) ∈ R2 : r0 < r < 1}
with the covering map
ϕ : Â → A;
ϕ(r , t) = reit .
I
Sarason develops a BSF factorization for modulus
automorphic functions Â
I
Unfortunately, these result don’t translate easily to H p (A)
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Positivity in the Disc
Positivity in the Annulus
Generalizations
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 the
set of zeros of f repeated according to multiplicity. Then
∞
X
n=1
r0
min 1 − |an |, 1 −
|an |
Jason Ekstrand
< ∞.
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Positivity in the Disc
Positivity in the Annulus
Generalizations
Theorem (Ekstrand, 2014)
Let f ∈ H ∞ (A) that is not identically zero and let {an } be the roots of f
repeated according to multiplicity. Then the Blaschke products
B1 (z) =
Y
√
|an |≥ r0
ān an − z
|an | 1 − ān z
and
B2 (z) =
Y
√
|an |< r0
an r0 /an − z
|an | 1 − (r0 /ān )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 a
continuous extension to A− then so does g.
Theorem (Ekstrand, 2014)
An element f ∈ H p (A) is positive if and only if f = g ∗ g for some
g ∈ H 2p (A). Furthermore, if f is continuous on A− , then g may
be chosen continuous on A− .
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Positivity in the Disc
Positivity in the Annulus
Generalizations
Generalizations to other domains
Definition
Let G be a domain. We say that G is symmetric if
G = G∗ = {z̄ : z ∈ G}.
Theorem (Ekstrand, 2014)
Let G be a symmetric domain where ∂G is the union of finitely
many disjoint Jordan curves and let f ∈ H ∞ (G). Then f is
positive 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).
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Connections with Representation Theory
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, ∗) are
exactly the point-evaluations on G ∩ R.
I
f ∈ A(G, ∗) is positive if and only if ϕ(f ) ≥ 0 for every
one-diimensional ∗-representation ϕ of A(G, ∗).
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Theorem (Ekstrand, 2014)
Let G be a region so that ∂G is the union of finitely many
disjoint 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,
P
3. f = ni=1 gi∗ 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
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Future Work
1. Extend the results to even more general domains
I
I
While the restriction that ∂G is the union of finitely many
disjoint Jordan curves is sufficient, I have no proof that it is
necessary.
Unfortunately, such an extension would probably need a
new technique.
2. Try and extend these results to a non-abelian case
I
These definitions extend fairly easily to Mn×n (A(G))
3. Consider domains not in C such as Riemann surfaces
I
There is a 1965 paper by Voichick and Zalcman that gives a
BSF factorization for a certain class of Riemann surfaces
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
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, London
Mathematical Society Student Texts, vol. 39, Cambridge University
Press, Cambridge, 1997.
[4] John B. Conway, A course in functional analysis, 2nd ed., Graduate
Texts 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.
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
References II
[7] John B. Conway, Functions of one complex variable, 2nd ed., Graduate
Texts in Mathematics, vol. 11, Springer-Verlag, New York-Berlin, 1978.
[8] Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book
Co., New York, 1987.
[9] Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall
Series in Modern Analysis, Prentice-Hall Inc., Englewood Cliffs, N. J.,
1962.
[10] Donald Sarason, The H p spaces of an annulus, Mem. Amer. Math. Soc.
No. 56 (1965), 78.
[11] Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren
der Mathematischen Wissenschaften [Fundamental Principles of
Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992.
[12] Rolf Nevanlinna, Analytic functions, Translated from the second German
edition by Phillip Emig. Die Grundlehren der mathematischen
Wissenschaften, Band 162, Springer-Verlag, New York, 1970.
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
References III
[13] Michael Voichick and Lawrence Zalcman, Inner and outer functions on
Riemann surfaces, Proc. Amer. Math. Soc. 16 (1965), 1200–1204.
[14] Stewart S. Cairns, An elementary proof of the Jordan-Schoenflies
theorem, Proc. Amer. Math. Soc. 2 (1951), 860–867.
[15] Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren
der Mathematischen Wissenschaften [Fundamental Principles of
Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992.
[16] Ryuji Maehara, The Jordan curve theorem via the Brouwer fixed point
theorem, Amer. Math. Monthly 91 (1984), no. 10, 641–643, DOI
10.2307/2323369.
Thank You!
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Idea Behind the Proof
Theorem (Riemann Mapping Theorem)
Let G ⊆ C be a simply connected region that is not the whole
plane and let a ∈ G. Then there is a unique holomorphic
bijection φ : G → D so that φ(a) = 0 and φ0 (a) > 0.
Theorem (Carathéodory)
Let G ⊆ C be a simply connected region whose boundary is a
Jordan curve. Then the Riemann map φ : G → D extends to a
homeomorphism Φ : G− → D− .
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Sketch of Proof
Start with some symmetric region G and f ∈ H ∞ (G)
G
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Sketch of Proof
Pick a single hole H in G
G
H
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Sketch of Proof
Define a Carathéodory map φ : C \ H → D−
φ
φ(H)
G
H
φ(G)
G
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
Sketch of Proof
Pick r0 so that {z ∈ C : r0 ≤ |z| < 1} ⊆ φ(G)
φ
φ(H)
G
H
φ(G)
G
Jason Ekstrand
Positivity in Function Algebras
Introduction
Positivity
Connections with Representation Theory
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 Carathéodory theorem, φ is a
homeomorphism of C \ H and D− so continuity follows
from results in the annulus.
Jason Ekstrand
Positivity in Function Algebras