Category Archives: Functional analysis

Another one bites the dust (actually many of them)

Functional analysis, Open problem, Operator algebras, Operator theory

Boom. In the arxiv mailing list of a few days ago appeared the following paper: “Interlacing Families II: Mixed Characteristic Polynomials and The Kadison-Singer Problem” (Markus, Spielman and Srivastava). The abstract says:

We use the method of interlacing families of polynomials to prove Weaver’s conjecture KS2, which is known to imply a positive solution to the Kadison-Singer problem via a projection paving conjecture of Akemann and Anderson. Our proof goes through an analysis of the largest roots of a family of polynomials that we call the “mixed characteristic polynomials” of a collection of matrices.

From the abstract it might not be immediately clear that this paper claims to solve the Kadison-Singer problem, because it says that their result implies KS via another conjecture; what they mean, however, is that the conjecture they prove was proven to be equivalent to another conjecture which has already been shown in the past to be equivalent to a positive solution to the Kadison-Singer problem.

Blog posts on the solution appeared here and here, with links to excellent references. I will add here a few remarks of my own.

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Forty five years later, a major open problem in operator algebras is solved

Arveson, Expository, Operator algebras

A couple of days ago, Ken Davidson and Matt Kennedy posted a preprint on the arxiv, “The Choquet boundary of an operator system“. In this paper they solve a major open problem in operator algebras, showing that every operator system has sufficiently many boundary representations. 

In 1969, William Arveson published the seminal paper, [“Subalgebras of C*-algebras”, Acta Math. 123, 1969], which is one of the cornerstones, (if not the cornerstone) of the theory of operator spaces and nonself-adjoint operator algebras. In that paper, among other things, Arveson introduced and put to good use the notion of a boundary representation. I wrote on “Subalgebras of C*-algebras” in a previous post dedicated to Arveson, and for some background material the reader is invited to look into that old post. I did not, however, write much about boundary representations (because I was emphasizing his contributions rather what he has left open). Below I wish to explain what are boundary representations, what does it mean that there are sufficiently many of these, and where Davidson and Kennedy’s new results fits in the chain of results leading to the solution of the problem. The paper itself is accessible to anyone who understands the problem, and the main ideas are clearly presented in its introduction.

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

Advanced Analysis, 20125401, Complex variables, Functional analysis, Operator theory, ,

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 $latex z_1, ldots, z_n in D$, and $latex w_1, ldots, w_n in mathbb{C}$ be given. There exists a function $latex f in H^infty(D)$ satisfying $latex |f|_infty leq 1$ and 

$latex f(z_i) = w_i ,, ,, i=1, ldots, n$

if and only if the following matrix inequality holds:

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

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

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

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).

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

Souvenirs from Bangalore

Conference, Operator algebras, Operator theory, Research, , , ,

I recently returned from the two week long workshop and conference Recent Advances in Operator Theory and Operator Algebras which took place in ISI Bangalore. As I promised myself before going, I was on the look-out for something new to be excited about and to learn. The event (beautifully organized and run) was made of two parts: a workshop, which was a one week mini-school on several topics (see here for topics) and a one week conference. It was very very broad, and there were several talks (or informal discussions) which I plan to pursue further.

In this post and also perhaps in a future one I will try to work out (for my own benefit, mostly) some details of a small part of the research presented in two of the talks. The first part is the Superproduct Systems which arise in the theory of E_0-semigroups on type II_1 factors (following the talk of R. Srinivasan). The second (which I will not discuss here, but perhpas in the future) is the equivalence between the Baby Corona Theorem and the Full Corona Theorem (following the mini-course given by B. Wick). In neither case will I describe the most important aspect of the work, but something that I felt was urgent for me to learn. 

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

Advanced Analysis, 20125401, Functional analysis, Operator algebras,

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))

Advanced Analysis, 20125401, Functional analysis, ,

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)

Advanced Analysis, 20125401, Functional analysis, ,

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 11: Banach spaces (weak topologies, Alaoglu’s theorem)

Advanced Analysis, 20125401, Functional analysis

Let $latex X$ be the Banach space $latex C([0,1])$ of continuous functions on the interval $latex [0,1]$ with the sup norm. Consider the following sequence of functions $latex {f_n}$ defind as follows. $latex f(0) = 0 $ and $latex f_n(1/(n+1)) = 1$ for all $latex n = 1, 2, ldots$,  $latex f_n$ is equal to zero on the interval between $latex 2/(n+1)$ and $latex 1$, and $latex f_n$ is linear in the intervals where we haven’t defined it yet (visualize!). The sequence is tending to zero pointwise, but the norm of $latex X$ does not detect this. The sequence tends to $latex 0$ in the $latex L^1$ norm, but the $latex L^1$ norm is not in the game. Can the Banach space structure of $latex X$ detect the convergence of $latex f_n$ to $latex 0$? Continue reading Advanced Analysis, Notes 11: Banach spaces (weak topologies, Alaoglu’s theorem) →

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

Advanced Analysis, 20125401, Functional analysis,

Recall Theorem 6 from Notes 3:

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

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.  Continue reading Advanced Analysis, Notes 10: Banach spaces (application: divergence of Fourier series) →

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

Advanced Analysis, 20125401, Functional analysis

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“.  Continue reading Advanced Analysis, Notes 9: Banach spaces (the three big theorems) →