## Month: November, 2012

### 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.

### 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.

### 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.

I have been writing my own lecture notes (based, of course, on an array of well known references and also on some of my notes from my graduate studies) for the course Advanced Analysis. I have decided to do this because, on the one hand, there was no particular text that I wanted to follow, while on the other hand I wanted my students to have a convenient reference for the material in the course. I hope these notes were instructive to some readers of the blog.

During the second half of the course I will follows Arveson’s book “A Short Course on Spectral Theory”, besides some applications and/or examples that I will want to occasionally throw in. So I will post my notes on much rarer occasions.

### Advanced Analysis, Notes 11: Banach spaces (weak topologies, Alaoglu’s theorem)

Let $X$ be the Banach space $C([0,1])$ of continuous functions on the interval $[0,1]$ with the sup norm. Consider the following sequence of functions $\{f_n\}$ defind as follows. $f(0) = 0$ and $f_n(1/(n+1)) = 1$ for all $n = 1, 2, \ldots$,  $f_n$ is equal to zero on the interval between $2/(n+1)$ and $1$, and $f_n$ is linear in the intervals where we haven’t defined it yet (visualize!). The sequence is tending to zero pointwise, but the norm of $X$ does not detect this. The sequence tends to $0$ in the $L^1$ norm, but the $L^1$ norm is not in the game. Can the Banach space structure of $X$ detect the convergence of $f_n$ to $0$? Read the rest of this entry »

### Advanced Analysis, Notes 10: Banach spaces (application: divergence of Fourier series)

Recall Theorem 6 from Notes 3:

Theorem 6: For every $f \in C_{per}([0,1]) \cap C^1([0,1])$, the Fourier series of $f$ converges uniformly to $f$

It is natural to ask how much can we weaken the assumptions of the theorem and still have uniform convergence, or how much can we weaken and still have pointwise convergence. Does the Fourier series of a continuous (and periodic) function always converge? In this post we will use the principle of uniform boundedness to see that the answer to this question is a very big NO.

Once again, we begin with some analytical preparations.  Read the rest of this entry »

### Advanced Analysis, Notes 9: Banach spaces (the three big theorems)

Until now we had not yet seen a theorem about Banach spaces — the Hahn–Banach theorems did not require the space to be complete. In this post we learn the three big theorems about operators on Banach spaces: the principle of uniform boundedness, the open mapping theorem, and the closed graph theorem. It is common that these three theorems are presented in texts on functional analysis under the heading “consequences of the Baire category theorem“.  Read the rest of this entry »

### On the isomorphism question for complete Pick multiplier algebras

In this post I want to tell you about our new preprint, “On the isomorphism question for complete Pick multiplier algebras“,  which Matt Kerr, John McCarthy and myself just uploaded to the arXiv. Very broadly speaking, the motif of this paper is the connection between algebra and geometry; to be a little bit more precise, it is the connection between complex geometry and nonself-adjoint operator algebras. Read the rest of this entry »

### Advanced Analysis, Notes 8: Banach spaces (application: weak solutions to PDEs)

Today I will show you an application of the Hahn-Banach Theorem to partial differential equations (PDEs). I learned this application in a seminar in functional analysis, run by Haim Brezis, that I was fortunate to attend in the spring of 2008 at the Technion.

As often happens with serious applications of functional analysis, there is some preparatory material to go over, namely, weak solutions to PDEs.

### Advanced Analysis, Notes 7: Banach spaces (dual spaces and duality, Lp spaces, the double dual, quotient spaces)

Today we continue our treatment of the dual space $X^*$ of a normed space (usually Banach) $X$. We start by considering a wide class of Banach spaces and their duals.  Read the rest of this entry »