Noncommutative Analysis

Tag: corrigendum

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).

 

A corrigendum

Matt Kennedy and I have recently written a corrigendum to our paper “Essential normality, essential norms and hyperrigidity“. Here is a link to the corrigendum. Below I briefly explain the gap that this corrigendum fills.

A corrigendum is correction to an already published paper. It is clear why such a mechanism exists: we want the papers we read to represent true facts, so false claims, as well as invalid proofs or subtle gaps should be pointed out to the community. Now, many many papers (I don’t want to say “most”) have some kind of mistake in them, but not every mistake deserves a corrigendum – for example there are mistakes that the reader will easily spot and fix, or some where the reader may not spot the mistake, but the fix is simple enough.

There are no rules as to what kind of errors require a corrigendum. This depends, among other things, on the authors. Some mistakes are corrected by other papers. I believe that very quickly some sort of mechanism – say google scholar, or mathscinet – will be able to tell if the paper you are looking up is referenced by another paper pointing out a gap, so such a correction-in-another-paper may sometimes serve as legitimate replacement for a corrigendum, when the issue is a gap or minor mistake.

There is also a question of why publish a corrigendum at all, instead of updating the version of the paper on the arxiv (and this is exactly what the moderators of the arxiv told us at first when we tried to upload our corrigendum there. In the end we convinced them that the corrigendum can stand by itself). I think that once a paper is published, it could be confusing to have a version more advanced than the published version; it becomes very clumsy to cite papers like that.

The paper I am writing about (see this post to see what its about) had a very annoying gap: we justified a certain step by citing a particular proposition from a monograph. The annoying part is that the proposition we cite does not exactly deal with the situation we deal with in the paper, but our idea was that the same proof works in our situation. We did not want to spell out the details because we considered that to be very easy, and in any case it was not a new argument. Unfortunately, the same proof does work when working with homogeneous ideals (which was what first versions of the paper treated) but in fact it is not clear if they work for non-homogeneous ideals. The reason why this gap is so annoying, is that it leads the reader to waste time in a wild goose chase: first the reader goes and finds the monograph we cite, looks up the result (has to read also a few extra pages to see he understands the setting and notation in the monograph), realises this is is not the same situation, then tries to adapt the method but fails. A waste of time!

Another problem that we had in our paper is that one requires our ideals to be “sufficiently non-trivial”. If this were the only problem we would perhaps not bother writing a corrigendum just to introduce a non-triviality assumption, since any serious reader will see that we require this.

If I try to take a lesson from this, besides a general “be careful”, it is that it is dangerous to change the scope of the paper (for us – moving form homogeneous to non-homogeous ideals) in late stages of the preparation of the paper. Indeed we checked that all the arguments work for the non-homogneous case, but we missed the fact that an omitted argument did not work.

Our new corrigendum is detailed and explains the mathematical problem and its solutions well, anyone seriously interested in our paper should look at it. The bottom line is this as follows.

Our paper has two main results regarding quotients of the Drury-Arveson module by a polynomial ideal. The first is that the essential norm in the non selfadjoint algebra associated to a the quotient module, as well as the C*-envelope, are as the Arveson conjecture predicts (Section 3 in the paper) . The second is that essential normality is equivalent to hyperrigidity (Section 4 in the paper).

Under the assumption that all our ideals are sufficiently non-trivial (and some other standing assumptions stated in the paper), the situation is as follows.

The first result holds true as stated.

For the second result, we have that hyperrigidity implies essential normality (as we stated), but the implication “essential normality implies hyperrigidity” is obtained for homogeneous ideals only.