Arveson’s hyperrigidity conjecture refuted by Bilich and Dor-On

Boom! This morning Boris Bilich and Adam Dor-On published a short preprint on the arXiv “Arveson’s hyperrigidity conjecture is false” in which they provide a counter example that refutes Arveson’s hyperrigidity conjecture. This is a fantastic achievement! It is one of the most interesting things that happened in my field lately and also somewhat of a surprise, a paper that is sure to make a significant impact on the subject.

(I should say that Adam was kind enough to let me read the manuscript a week ago, so that I had time already to check the details and as far as I can tell it looks correct.)

Let us recall quickly what the conjecture is (for more background see the series of posts that I wrote for the topics course I gave several years ago).

Let be a unital operator algebra generating a C*-algebra .

Aleman, Hartz, McCarthy and Richter characterize interpolating sequences in complete Pick spaces

The purpose of this post is to discuss the recent important contribution by Aleman, Hartz, McCarthy and Richter to the characterization of interpolating sequences (for multiplier algebras of certain Hilbert function spaces). Their recent paper “Interpolating sequences in spaces with the complete Pick property” was uploaded to the arxiv about two weeks ago; here I will just give some background and state the main result. (Even more recently these four authors released yet another paper that looks very interesting – this one.)

1. Background – interpolating sequences

We will be working with the notion of Hilbert function spaces – also called reproducing Hilbert spaces (see this post for an introduction). Suppose that is a Hilbert function space on a set , and its reproducing kernel. The Pick interpolation problem is the following:

One of the most outrageous open problems in operator/matrix theory is solved!

I want to report on a very exciting development in operator/matrix theory: the von Neumann inequality for matrices has been shown to hold true. I learned this from a recent paper (with the irresistible title) “The von Neumann inequality for matrices“, posted on the arxiv by Greg Knese. In this paper, Knese explains how the solution of this outstanding open problem follows from results in a paper by Lukasz Kosinski, “The three point Nevanlinna-Pick problem in the polydisc” that appeared on the arxiv about a half a year ago. Beautifully, and not surprisingly, the solution of this operator/matrix theoretic problem follows from deep new facts in complex function theory in several variables.

To recall the problem, let us denote the operator norm of a matrix , and for every polynomial in variables we denote by the supremum norm

.

A matrix is said to be a contraction (or contractive) if .

We say that commuting contractions  satisfy von Neumann’s inequality if 

(*)  

for every polynomial in variables.

It was known since the 1960s that (*) holds when . Moreover, it was known that for , there are counter examples, consisting of contractive matrices that do not satisfy von Neumann’s inequality. On the other hand, it was known that (*) holds for any if the matrices are of size . Thus, the only missing piece of information was whether or not von Neumann’s inequality holds or not for three or more contractive  matrices. To stress the point: it was not known whether or not von Neumann’s inequality holds for three three-by-three matrices. The problem in this form has been open for 15 years  – but the problem is much older: in 1974 Kaiser and Varopoulos came up with a counter-example, and since then both the  and the cases were open until Holbrook in 2001 found a counter example. You have to agree that this is outrageous, perhaps even ridiculous, I mean, three matrices, come on!

In Knese’s paper this story and the positive solution to the problem is explained very clearly and succinctly, and is recommended reading for any operator theorist. One has to take on faith the paper of Kosinski which, as Knese stresses, is where the major new technical advance has been made (though one should not over-stress this fact, because tying things together, the way Knese has done, requires a deep understanding of this problem and of the various ingredients). To understand Kosinki’s paper would require a greater investment of time, but it appears that the paper has already been accepted for publication, so I am quite confident and happy to see this problem go down.

Daniel Spielman talks at HUJI – thoughts

I got an announcement in the email about the “Erdos Lectures”, that will be given by Daniel Spielman in the Hebrew University of Jerusalem next week (here is the poster on Gil Kalai’s blog). The title of the first lecture is “The solution of the Kadison-Singer problem”. Recall that not long ago Markus, Spielman and Srivastava proved Weaver’s KS2 conjecture, which implies a positive solution to Kadison-Singer (the full story been worked out to expository perfection on Tao’s blog).

My immediate response to this invitation was to start planning a trip to Jerusalem on Monday – after all it is not that far, it’s about a solution of a decades old problem, and Daniel Spielman is sort of a Fields medalist. I highly recommend to everyone to go hear great scientists live whenever they have the opportunity. At worst, their lectures are “just” inspiring. It is not for the mathematics that one goes for in these talks, but for all the stuff that goes around mathematics (George Mostow’s unusual colloquium given at BGU on May 2013 comes to mind).

But then I remembered that I have some obligations on Monday, so I searched and found a lecture by Daniel Spielman with the same title online: here. Watching the slides with Spielman’s voice is not as inspiring as hearing and seeing a great mathematician live, but quite good. He makes it look so easy!

In fact, Spielman does not discuss KS at all. He says (about a minute into the talk) “Actually, I don’t understand, really, the Kadison-Singer problem”. A minute later he has a slide where the problem is written down, but he says “let me not explain what it is”, and sends the audience to read Nick Harvey’s survey paper (which is indeed very nice). These were off-hand remarks, and I should not catch someone at his spoken word, (and I am sure that even things that Spielman would humbly claim to “not understand, really”, he probably understands as well as I do, at least), but the naturality in which the KS problem was pushed aside in a talk about KS made we wonder.

In the post I put up soon after appearance of the paper I wrote (referring to the new proof of KS2) that “… this looks like a very nice celebration of the Unity of Mathematics”. I think that in a sense the opposite is also true. I will try to reformulate what I wrote.

“The solution of KS is a beautiful and intriguing manifestation of the chaotic, sticky, psychedelic, thickly interwoven, tangled, scattered, shattered and diffuse structure of today’s mathematics.”

I don’t mean that in a bad way. I mean that a bunch of deep conjectures, from different fields, most of which, I am guessing, MSS were not worried about, were shown over several decades to be equivalent to each other, and were ultimately reduced (by Weaver) to a problem on the arrangement of vectors in finite dimensional spaces (Discrepancy Theory), and eventually solved, following years of hard work, by three brilliant mathematicians using ingenious yet mostly elementary tools. The problem solved is indeed interesting in itself, and the proof is also very interesting, but it seems that the connection with “Kadison-Singer” is more a trophy than a true reward.

It would be very interesting now to think of all the equivalent formulations with hindsight, and seek the unifying structure, and to try to glean a reward.

 

 

Major advances in the operator amenability problem

Laurent Marcoux and Alexey Popov recently published a preprint, whose title speaks for itself :”Abelian, amenable operator algebras are similar to C*-algebras“. This complements another recent contribution, by Yemon Choi, Ilijas Farah and Narutaka Ozawa, “A nonseparable amenable operator algebra which is not isomorphic to a C*-algebra“.

The open problem that these two papers address is whether every amenable Banach algebra, which is a subalgebra of , is similar to a (nuclear) C*-algebra. As the titles clearly indicate (good titling!), we now know that an abelian amenable operator algebra is similar to a C*-algebra, and on the other hand, that a non-separable, non-abelian operator algebra is not necessarily similar to a C*-algebra.

I recommend reading the introduction to the Marcoux-Popov paper (which is very friendly to non-experts too) to get a picture of this problem, its history, and an outline of the solution.