Noncommutative Analysis

Tag: matrix range

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.

The complex matrix cube problem – new results from summer projects

In this post I will summarize the results obtained by my group in the “Summer Projects Week” that took place two weeks ago at the Technion. As in last time (see here for a summary of last year’s project) the title of the project I suggested was “Numerical Explorations of Open Problems from Operator Theory”. This time, I was lucky to have Malte Gerhold and Satish Pandey, my postdocs, volunteer to help me with the mentoring. The students who chose our project were Matan Gibson and Ofer Israelov, and they did fantastic work.

Read the rest of this entry »

New paper “Compressions of compact tuples”, and announcement of mistake (and correction) in old paper “Dilations, inclusions of matrix convex sets, and completely positive maps”

Ben Passer and I have recently uploaded our preprint “Compressions of compact tuples” to the arxiv. In this paper we continue to study matrix ranges, and in particular matrix ranges of compact tuples. Recall that the matrix range of a tuple A = (A_1, \ldots, A_d) \in B(H)^d is the the free set \mathcal{W}(A) = \sqcup_{n=1}^\infty \mathcal{W}_n(A), where

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

A tuple A is said to be minimal if there is no proper reducing subspace G \subset H such that \mathcal{W}(P_G A\big|_G) = \mathcal{W}(A). It is said to be fully compressed if there is no proper subspace whatsoever G \subset H such that \mathcal{W}(P_G A\big|_G) = \mathcal{W}(A).

In an earlier paper (“Dilations, inclusions of matrix convex sets, and completely positive maps”) I wrote with other co-authors, we claimed that if two compact tuples A and B are minimal and have the same matrix range, then A is unitarily equivalent to B; see Section 6 there (the printed version corresponds to version 2 of the paper on arxiv). This is false, as subsequent examples by Ben Passer showed (see this paper). A couple of other statements in that section are also incorrect, most obviously the claim that every compact tuple can be compressed to a minimal compact tuple with the same matrix range. All the problems with Section 6 of that earlier paper “Dilations,…” can be quickly  fixed by throwing in a “non-singularity” assumption, and we posted a corrected version on the arxiv. (The results of Section 6 there do not affect the rest of the results in the paper, and are somewhat not in the direction of the main parts of that paper).

In the current paper, Ben and I take a closer look at the non-singularity assumption that was introduced in the corrected version of “Dilations,…”, and we give a complete characterization of non-singular tuples of compacts. This characterization involves the various kinds of extreme points of the matrix range \mathcal{W}(A). We also make a serious invetigation into fully compressed tuples defined above. We find that a matrix tuple is fully compressed if and only if it is non-singular and minimal. Consequently, we get a clean statement of the classification theorem for compacts: if two tuples A and B of compacts are fully compressed, then they are unitarily equivalent if and only if \mathcal{W}(A) = \mathcal{W}(B).