Noncommutative Analysis

Category: Research

Seminar talk

Next Tuesday, May 19th, at 14:30 (Israeli time), I will give a video talk at the Séminaire d’Analyse Fonctionnelle “in” Laboratoire de mathématiques de Besançon. It will be about my recent paper with Michael Skeide, the one that I announced here.

Title: CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: the Multi-Parameter Case and Beyond.


Abstract: We introduce a framework for studying dilations of semigroups of completely positive maps on von Neumann algebras. The heart of our method is the systematic use of families of Hilbert C*-correspondences that behave nicely with respect to tensor products: these are product systems, subproduct systems and superproduct systems. Although we developed our tools with the goal of understanding the multi-parameter case, they also lead to new results even in the well studied one parameter case. In my talk I will give a broad outline and a taste of the dividends our work.

The talk is based on a recent joint work with Michael Skeide.

Assumed knowledge: Completely positive maps and C*-algebras.

Feel free to write to me if you are interested in a link to the video talk.

Dilations of q-commuting unitaries

Malte Gerhold and I just have just uploaded a revision of our paper “Dilations of q-commuting unitaries” to the arxiv. This paper has been recently accepted to appear in IMRN, and was previously rejected by CMP, so we have four anonymous referees and two handling editors to be thankful to for various corrections and suggested improvements (though, as you may understand, one editor and two referees have reached quite a wrong conclusion regarding our beautiful paper :-).

This is a quite short paper (200 full pages shorter than the paper I recently announced), which tells a simple and interesting story: we find that optimal constant c_\theta, such that every pair of unitaries u,v satisfying the q-commutation relation

vu = e^{i\theta} uv

dilates to a pair of commuting normal operators with norm less than or equal to c_\theta (this problems is related to the “complex matrix cube problem” that we considered in the summer project half year ago and the one before). We provide a full solution. There are a few ramifications of this idea, as well as surprising connections and applications, so I invite you to check out the nice little introduction.

CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: the Multi-Parameter Case and Beyond

Reblogging the announcement of my recent giant paper with Michael Skeide. I want this to be at the head of my blog for a while.

Noncommutative Analysis

Michael Skeide and I have recently uploaded our new paper to the arxiv: CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: The Multi-Parameter Case and Beyond. In this gigantic (219 pages) paper, we propose a framework for studying the dilation theory of CP-semigroups parametrized by rather general monoids (i.e., semigroups with unit), and we use this framework for obtaining new results regarding the possibility or impossibility of constructing or having a dilation, we use it also for obtaining new structural results on the “mechanics” of dilations, and we analyze many examples using our tools. We present results that we have announced long ago, as well as some surprising discoveries.

This is an exciting moment for me, since we have been working on this project for more than a decade.

“Excuse me, did you really say decade?”

View original post 2,653 more words

CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: the Multi-Parameter Case and Beyond

Michael Skeide and I have recently uploaded our new paper to the arxiv: CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: The Multi-Parameter Case and Beyond. In this gigantic (219 pages) paper, we propose a framework for studying the dilation theory of CP-semigroups parametrized by rather general monoids (i.e., semigroups with unit), and we use this framework for obtaining new results regarding the possibility or impossibility of constructing or having a dilation, we use it also for obtaining new structural results on the “mechanics” of dilations, and we analyze many examples using our tools. We present results that we have announced long ago, as well as some surprising discoveries.

This is an exciting moment for me, since we have been working on this project for more than a decade.

“Excuse me, did you really say decade?”

Read the rest of this entry »

New paper: “On the matrix range of random matrices”

Malte Gerhold and I recently posted our new paper “On the matrix range of random matrices” on the arxiv, and I want to write a few words about it.

Recall that the matrix range of a d-tuple of operators A = (A_1, \ldots, A_d) \in B(H)^d is the noncommutative set W(A) = \cup_n W_n(A), where

W_n(A) = \{ (\phi(A_1), \ldots, \phi(A_d)) : \phi : B(H) \to M_n is UCP \}.

The matrix range appeared in several recent papers of mine (for example this one), it is a complete invariant for the unital operator space generated by A_1 \ldots, A_d, and is within some classes is also a unitary invariant.

The idea for this paper came from my recent (last couple of years or so) flirt with numerical experiments. It has dawned on me that choosing matrices randomly from some ensembles, for example by setting

G = randn(N);

X = (G + G')/sqrt(2*N);

(this is the GOE ensemble) is a rather bad way for testing “classical” conjectures in mathematics, such as what is the best constant for some inequality. Rather, as N increases, random N \times N behave in a very “structured” way (as least in some sense). So we were driven to try to understand, roughly what kind of operator theoretic phenomena do we tend to observe when choosing random matrices.

The above paragraph is a confession of the origin of our motive, but at the end of the day we ask and answer honest mathematical questions with theorems and proofs. If X^N = (X^N_1, \ldots, X^N_d) is a d-tuple of N \times N matrices picked at random according to the Matlab code above, then experience with the law of large numbers, the central limit theorem, and Wigner’s semicircle law, suggests that W(X^N) will “converge” to something. And by experience with free probability theory, if it converges to something, then is should be the matrix range of the free semicircular tuple. We find that this is indeed what happens.

Theorem: Let X^N be as above, and let s = (s_1, \ldots, s_d) be a semicircular family. Then for all n,

\lim_{N \to \infty} d_H(W_n(X^N),W(s)) = 0 almost surely

in the Hausdorff metric.

The semicircular tuple s = (s_1, \ldots, s_d) is a certain d-tuple of operators that can be explicitly described (see our paper, for example).

We make heavy use of some fantastic results in free probability and random matrix theory, and our contribution boils down to finding the way to use existing results in order to understand what happens at the level of matrix ranges. This involves studying the continuity of matrix ranges for continuous fields of operators, in particular, we study the relationship between the convergence

(*) \lim_{N \to \infty} \|p(X^N)\| = \|p(X)\|

(which holds for X^N as above and X = s by a result of Haagerup and Torbjornsen) and

(**) \lim_{N \to \infty} d_H(W_n(X^N),W(X)) = 0.

To move from (*) to (**), we needed to devise a certain quantitative Effros-Winkler Hahn-banach type separation theorem for matrix convex sets.