## Category: Functional analysis

### Topics in Operator Theory, Lecture 10: hyperrigidity

In this lecture we discuss the notion of hyperrigidity, which was introduced by Arveson in his paper The noncommutative Choquet boundary II: Hyperrigidity, shortly after he proved the existence of boundary representations (and hence the C*-envelope) for separable operator systems. Most of the results and the examples that we will discuss in this lecture come from that paper, and we will certainly not be able to cover everything in that paper. In the last section of this post I will put some links concerning a result of Kennedy and myself which connects hyperrigidity to the Arveson’s essential normality conjecture.

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### Topics in Operator Theory, Lecture 9: the boundary theorem

In this post, we come back to boundary representations and the C*-envelope, prove an important theorem, and see some examples. It is interesting to note that the theory has interesting consequences even for operators on finite dimensional spaces. Here is a link to a very interesting paper by Farenick giving an exposition of Arveson’s boundary theorem in the setting of operators on finite dimensional spaces.

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### Topics in Operator Theory, Lecture 8: matrix convexity

In this lecture we will encounter the notion of matrix convexity. Matrix convexity is an active area of research today, and an important tool in noncommutative analysis. We will define matrix convex sets, and we will see that closed matrix convex sets have matrix extreme points which play a role similar to extreme points in analysis. As an example of a matrix convex set, we will study the set of all matrix states. We will use these notions to outline the proof that there are sufficiently many pure UCP maps, something that was left open from the previous lecture.

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### The never ending paper

My paper On operator algebras associated with monomial ideals, written jointly with Evgenios Kakariadis, has recently appeared in Journal of Mathematical Analysis and Applications. They gave me a link to share (the link will work for the next several weeks): click here for an official version of the paper.

The paper is a very long paper, so it has a very long introduction too. To help to get into the heart of editors and referees, we wrote, at some point, a shorter cover letter which attempts to briefly explain what the main achievements are. See below the fold for that.

But first, a rant!

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### 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 $\mathcal{W}(A) = \mathcal{W}(B)$.