Noncommutative Analysis

Month: September, 2018

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).

 

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The complex matrix cube problem summer project – summary of results

In the previous post I announced the project that I was going to supervise in the Summer Projects in Mathematics week at the Technion. In this post I wish to share what we did and what we found in that week.

I had the privilege to work with two very bright students who have recently finished their undergraduate studies: Mattya Ben-Efraim (from Bar-Ilan University) and Yuval Yifrach (from the Technion). It is remarkable the amount of stuff they learned for this one week project (the basics of C*-algebras and operator spaces), and that they actually helped settle the question that I raised to them.

I learned a lot of things in this project. First, I learned that my conjecture was false! I also learned and re-learned some programming abilities, and I learned something about the subtleties and limitations of numerical experimentation (I also learned something about how to supervise an undergraduate research project, but that’s besides the point right now).

Statement of the problem

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