Noncommutative Analysis

Category: Advanced Analysis, 20125401

K-spectral sets and the holomorphic functional calculus

In two previous posts I discussed the holomorphic functional calculus as part of a standard course in functional analysis (lectures notes 18 and 19). In this post I wish to discuss a slightly different approach, which relies also on the notion of K-spectral sets, and relies a little less on contour integration of Banach-space valued functions.

In my very personal opinion this approach is a little more natural then the standard one, and it would be even more natural if one was able to altogether remove the dependence on Banach-space valued integrals (unfortunately, right now I don’t know how to do this completely).

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Advanced Analysis, Notes 19: The holomorphic functional calculus II (definition and basic properties)

In this post we continue our discussion of the holomorphic functional calculus for elements of a Banach algebra (or operators). The beginning of this discussion can be found in Notes 18. Read the rest of this entry »

Advanced Analysis, Notes 18: The holomorphic functional calculus I (motivation, definition, line integrals of holomorphic Banach-space valued functions)

This course, Advanced Analysis, contains some lectures which I have not written up as posts. For the topic of Banach algebras and C*-algebras the lectures I give in class follow pretty closely Arveson’s presentation from “A Short Course in Spectral Theory” (except that we do more examples in class). But there is one topic  – the holomorphic functional calculus -for which I decided to take a slightly different route, and for the students’ reference I am writing up my point of view.

Throughout this lecture we fix a unital Banach algebra A. By “unital Banach algebra” we mean that A is a Banach algebra with normalised unit 1_A.  For a complex number t \in \mathbb{C} we write t for t \cdot 1_A; in particular 1 = 1_A.  The spectrum \sigma(a) of an element a \in A is the set

\sigma(a) = \{t \in \mathbb{C} : a- t \textrm{ is not invertible in } A\}.

The resolvent set of a, \rho(a), is defined to be the complement of the spectrum,

\rho(a) = \mathbb{C} \setminus \sigma(a).

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Some links and announcements

  • The course “Advanced Analysis” is over. The lecture notes (the part that I prepared) are available here. Comments are very welcome. I hope to teach this course again in the not too far future and complete the lecture notes (add notes on Banach and C*-algebras, spectral theory and Fredholm theory). The homework exercises are available here, at the bottom of the page (the webpage is in Hebrew but the exercises are in English). 
  • In April Ken Davidson will be visiting our department at BGU. On this occasion we will hold a short conference, dates: April 9-10. Here is the conference page. Contact me for more details.
  • There are some interesting discussions going on in Gowers’s Weblog (see “Why I’ve joined the bad guys” and “Why I’ve joined the good guys” and some of the comments), regarding journals, publishing, new ideas, APCs, and so forth. The big news is that Gowers (after he kind of admits that being an editor of Forum of Mathematics makes him one of the bad guys) is now connected to another publishing adventure, that of epijournals, or arxiv overlay journals, which makes him one of the good guys (Just to set things straight: I think Gowers is a good guy). BTW: Gowers makes it clear that the credit for this initiative does not belong to him but to others, see his post.
  • I promised myself to stop writing about this topic, but I guess I am still allowed to put a link to something that I wrote about this in the past. So here is a link to a letter (also other letters) I sent to Letters to the Editor of the Notices. It is a response to this article by Rob Kirby.

Advanced Analysis, Notes 17: Hilbert function spaces (Pick’s interpolation theorem)

In this final lecture we will give a proof of Pick’s interpolation theorem that is based on operator theory.

Theorem 1 (Pick’s interpolation theorem): Let z_1, \ldots, z_n \in D, and w_1, \ldots, w_n \in \mathbb{C} be given. There exists a function f \in H^\infty(D) satisfying \|f\|_\infty \leq 1 and 

f(z_i) = w_i \,\, \,\, i=1, \ldots, n

if and only if the following matrix inequality holds:

\big(\frac{1-w_i \overline{w_j}}{1 - z_i \overline{z_j}} \big)_{i,j=1}^n \geq 0 .

Note that the matrix element \frac{1-w_i\overline{w_j}}{1-z_i\overline{z_j}} appearing in the theorem is equal to (1-w_i \overline{w_j})k(z_i,z_j), where k(z,w) = \frac{1}{1-z \overline{w}} is the reproducing kernel for the Hardy space H^2 (this kernel is called the Szego kernel). Given z_1, \ldots, z_n, w_1, \ldots, w_n, the matrix

\big((1-w_i \overline{w_j})k(z_i,z_j)\big)_{i,j=1}^n

is called the Pick matrix, and it plays a central role in various interpolation problems on various spaces.

I learned this material from Agler and McCarthy’s monograph [AM], so the following is my adaptation of that source.

(A very interesting article by John McCarthy on Pick’s theorem can be found here).

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Advanced Analysis, Notes 16: Hilbert function spaces (basics)

In the final week of the semester we will study Hilbert function spaces (also known as reproducing kernel Hilbert spaces) with the goal of presenting an operator theoretic proof of the classical Pick interpolation theorem. Since time is limited I will present a somewhat unorthodox route, and ignore much of the beautiful function theory involved. BGU students who wish to learn more about this should consider taking Daniel Alpay’s course next semester. Let me also note the helpful lecture notes available from Vern Paulsen’s webpage and also this monograph by Jim Agler and John McCarthy (in this post and the next one I will refer to these as [P] and [AM] below).

(Not directly related to this post, but might be of some interest to students: there is an amusing discussion connected to earlier material in the course (convergence of Fourier series) here).

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Advanced Analysis, Notes 15: C*-algebras (square root)

This post contains some make–up material for the course Advanced Analysis. It is a theorem about the positive square root of a positive element in a C*-algebra which does not appear in the text book we are using. My improvisation for this in class came out kakha–kakha, so here is the clarification.

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Advanced Analysis, Notes 14: Banach spaces (application: the Stone–Weierstrass Theorem revisited; structure of C(K))

In this post we will use the Krein–Milman theorem together with the Hahn–Banach theorem to give another proof of the Stone–Weierstrass theorem. The proof we present does not make use of the classical Weierstrass approximation theorem, so we will have here an alternative proof of the classical theorem as well.

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Advanced Analysis, Notes 13: Banach spaces (convex hulls and the Krein–Milman theorem)

It would be strange to disappear for a week without explanations. This blog was not working for the past week because of the situation in Israel. The dedication in the beginning of the previous post had something to do with this, too. We are now back to work, with the modest hope that things will remain quiet until the end of the semester. We begin our last chapter in basic functional analysis, convexity and the Krein–Milman theorem.

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Advanced Analysis, Notes 12: Banach spaces (application: existence of Haar measure)

This post is dedicated to my number one follower for her fourteenth birthday,which I spoiled…

Let G be a compact abelian group. By this we mean that G is at once both an abelian group and a compact Hausdorff topological space, and that the group operations are continuous, meaning that g \mapsto g^{-1} is continuous on G and (g,h) \mapsto g+h is continuous as a map from G \times G to G. It is known that there exists a regular Borel measure \mu on G, called the Haar measure, which is non-negative, satisfies \mu(G) = 1, and is translation invariant:

\forall g \in G . \mu(g+ E) = \mu(E) ,

for every Borel set E \subseteq G. In fact, the Haar measure is known to exist in greater generality (G does not have to be commutative and if one allows \mu to be infinite then G can also be merely locally compact). The Haar measure is an indispensable tool in representation theory and in ergodic theory. In this post we will use the weak* compactness of the unit ball of the dual to give a slick proof of the existence of the Haar measure in the abelian compact case.

1. The Kakutani–Markov fixed point theorem

The following theorem can be stated and proved in greater generality with more–or–less the same proof as we present.

Theorem 1 (Kakutani–Markov fixed point theorem): Let X be a Banach space, and let \mathcal{F} be a commuting family of weak-* continuous linear maps on X^*. Suppose that C is a non-empty, weak*–compact and convex subset of X^* such that T(C) \subseteq C for all T \in \mathcal{F}. Then \mathcal{F} has a common fixed point in C, i.e., there is an x \in C such that Tx = x for all T \in \mathcal{F}

Proof: Let us first prove the theorem in the case where \mathcal{F} = \{T\}. Choose some y_0 \in C. Construct the averages

y_N = \frac{1}{N+1} \sum_{n=0}^{N} T^n y_0 .

Since C is convex and T(C) \subseteq C, we have that y_N \in C for all C. Since C is compact, the sequence \{y_N\}_{N=1}^\infty contains a subnet \{y_{N(\alpha)}\} that converges to some y \in C. To show that Ty = y, we will prove that \langle x, Ty-y \rangle = 0 for all x \in X.

Fix x \in X. Then x is continuous on X^* in the weak* topology, hence bounded on the weak*–compact set C, say |\langle x, c \rangle| \leq M_x for all c \in C. Now for any N,

|\langle x , Ty_N - y_N \rangle | = \frac{1}{N+1} |\langle x, T^{N+1} y_0 - y_0 \rangle| \leq \frac{2M_x}{N+1} .

Thus |\langle x, T y - y \rangle| = \lim_\alpha |\langle x, T y_{N(\alpha)} - y_{N(\alpha)} \rangle | = 0 (recall that part of the definition of subnet is that N(\alpha) goes to infinity with \alpha). That completes the proof for the case where \mathcal{F} is a singleton.

Now let \mathcal{F} be arbitrary, and for every finite F \in \mathcal{F}, denote A_F = \{c \in C : \forall T \in F . Tc = c\}. A_F is evidently closed. We will show that the family \{A_F : F \subseteq \mathcal{F} \textrm{ finite}\} has the finite intersection property. It will follow from compactness that there is some x \in \cap_F A_F, which will be the sought after fixed point. Now, A_{F_1} \cap A_{F_2} = A_{F_1 \cup F_2}, so it follows that we only have to show that every A_F is non-empty. This is done by induction on the cardinality |F| of F. If |F| = 1, then A_F \neq \emptyset by the first half of the proof. Suppose that A_F is not empty, and let T \in \mathcal{F}. Then for every y \in A_F, we have for all S \in F

STy = TSy = Ty .

Therefore T(A_F) \subseteq A_F, so by the first half of the proof there is some z \in A_{F} fixed under T. In other words, z \in A_{F \cup \{T\}}, so this set is not empty.

2. The existence of Haar measure for abelian compact groups

We can now prove the existence of Haar measure for compact abelian groups.

Theorem 2: Let G be a compact Hausdorff abelian group. Then there exists a Haar measure for G. That is, there is a regular Borel probability measure \mu on G that is translation invariant. 

Proof: For every g \in G, let L_g : C(G) \rightarrow C(G) be the translation operator given by  (L_g f)(h) = f(g-h). We will find a regular Borel probability measure \mu on G such that for all g \in G,

(*) \int f d \mu = \int L_g f d \mu .

The regularity of the measure together with Urysohn’s Lemma then implies that \mu satisfies \mu(g+E) = \mu(E) for all Borel E and all g \in G (this might be trickier than it first seems).

Consider the family \mathcal{F} = \{T_g : g \in G\} of operators on M(X) = C(X)^* given by T_g = L_g^*. Then by Exercise G in Notes 11 T_g is weak* continuous for all g. Moreover, for all f \in C(X), \nu \in M(X) and g,h \in G,

\langle T_g T_h \nu, f \rangle = \langle \nu, L_h L_g f \rangle = \langle \nu, L_g L_h f \rangle = \langle T_h T_g \nu , f \rangle .

Therefore \mathcal{F} is a commuting family. Now let C be the subset of M(X) consisting of all probability measures. Then it is easy to see that C is weak* closed and convex, and that \mathcal{F} leaves C invariant. By the Kakutani–Markov Theorem, \mathcal{F} has a fixed point \mu \in C. By definition of C, \mu is a regular Borel probability measure on G. By the fact that \mu is a fixed point for \mathcal{F} we have \langle T_g \mu , f \rangle = \langle \mu , f \rangle, which is just another way of writing (*). That completes the proof.

Exercise A: It may seem as if the same argument would give the existence of a translation invariant regular Borel probability measure on a locally compact Hausdorff space.

  1. Prove why the theorem fails for non–compact spaces.
  2. What part of the argument breaks down?
  3. Make sure you know why that part doesn’t break down in the compact case.