Noncommutative Analysis

Month: September, 2015

Something sweet for the new year

Tim Gowers recently announced the start of a new journal, “Discrete Analysis”. The sweet thing about this journal is that it is an arxiv overlay journal, meaning that the journal will act like most other elctronic journals with the difference that all it does in the end (after standard peer review and editorial decisions) is put up a link on its website to a certain version of the preprint on the arxiv. The costs are so low, that neither readers nor authors are supposed to pay. In the beginning, Cambridge University will cover the costs of this particular journal, and there are hopes that funding will be found later (of course, arxiv has to be funded as well, but this does not seem to incur additional costs on arxiv). The journal uses a platform called Scholastica (which does charge something, but relatively low – like $10 per paper) so they did not have to set up their webpage and deal with that kind of stuff.

The idea has been around for several years and there are several other platforms (some of which do not charge anything since they are publicly funded) for carrying journals like this: Episciences, Open Journals. It seems like analysis, and operator theory in particular, are a little behind in these initiatives (correct me if I am wrong). But I am not worried, this is a matter of time.

The news of the baby journal made me especially happy since leaders like Gowers and Tao have been previously involved with the creation of the bad-idea-author-pay-journals Forum of Mathematics (Pi and Sigma), and it is great that their stature is also harnessed for a decent journal (which also happens to have a a nice and reasonable name).

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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 3 \times 3 matrices has been shown to hold true. I learned this from a recent paper (with the irresistible title) “The von Neumann inequality for 3 \times 3 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 \|A\| the operator norm of a matrix A, and for every polynomial p in d variables we denote by \|p\|_\infty the supremum norm

\|p\|_\infty = \sup_{|z_i|\leq 1} |p(z_1, \ldots, z_d)|.

A matrix A is said to be contractive if \|A\| \leq 1.

We say that d commuting contractions A_1, \ldots, A_d satisfy von Neumann’s inequality if 

(*)  \|p(A_1,\ldots, A_d)\| \leq \|p\|_\infty.

It was known since the 1960s that (*) holds when d \leq 2. Moreover, it was known that for d \geq 3, there are counter examples, consisting of d contractive 4 \times 4 matrices that do not satisfy von Neumann’s inequality. On the other hand, it was known that (*) holds for any d if the matrices A_1, \ldots, A_d are of size 2 \times 2. Thus, the only missing piece of information was whether or not von Neumann’s inequality holds or not for three or more contractive 3 \times 3 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 5 \times 5 counter-example, and since then both the 3 \times 3  and the 4 \times 4 cases were open until Holbrook in 2001 found a 4 \times 4 counter example. You have to agree that this is outrageous, perhaps even ridiculous, I mean, three 3 \times 3 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.