Noncommutative Analysis

Category: Conference

Souvenirs from Bangalore 2015

Last week I attended the conference “Complex Geometry and Operator Theory” in Indian Statistical Institute, Bangalore. The conference was also an occasion to celebrate Gadadhar Misra‘s 60s birthday.

As usual for me in conferences, I played a game with myself in which my goal was to find the most interesting new thing I learned, and then follow up on it to some modest extent. Although every day of the three day conference had at least two excellent lectures that I enjoyed, I have to pick one or two things, so here goes.

1. Noncommutative geometric means

The most exciting new-thing-I-learned was something that I heard not in a lecture but rather in a conversation I had with Rajendra Bhatia in one of the generously long breaks.

A very nice exposition of what I will briefly discuss below appears in this expository paper of Bhatia and Holbrook.

The notion of arithmetic mean generalizes easily to matrices. If A,B are matrices, then we can define

M_a(A,B) = \frac{A+B}{2}.

When restricted to hermitian matrices, this mean has some expected properties of a mean. For example,

  1. M_a(A,B) = M_a(B,A),
  2. If A \leq B, then A \leq M_a(A,B) \leq B,
  3. M_a(A,B) is monotone in its variables.

A natural question – which one may ask simply out of curiosity – is whether the geometric mean (x,y) \mapsto \sqrt{xy} can also be generalized to pairs of positive definite matrices. One runs into problems immediately, since if A and B are positive definite, one cannot extract a “positive square root” from AB, since when A and B do not commute then their product AB need not be a positive matrix.

It turns out that one can define a geometric mean as follows. For two positive definite matrices A and B, define

(*) M_g(A,B) = A^{1/2} \sqrt{A^{-1/2} B A^{-1/2}} A^{1/2} .

Note that when A and B commute (equivalently, when they are scalars) then M_g(A,B) reduces to \sqrt{AB}, so this is indeed a generalisation of the geometric mean. Not less importantly, it has all the nice properties of a mean, in particular properties 1-3 above (it is not evident that it is symmetric (the first condition), but assuming that the other two properties follow readily).

Now suppose that one needs to consider the mean of more than two – say, three – matrices. The arithmetic mean generalises painlessly:

M_a(A,B,C) = \frac{A + B + C}{3}.

As for the geometric mean, there has not been found an appropriate algebraic expression that generalises equation (*) above. About a decade ago, Bhatia, Holbrook and (separately) Moakher, found a geometric way to define the geometric mean of any number of positive definite matrices.

They key is that they view the set \mathbb{P}_n of positive definite n \times n matrices as a Riemannian manifold, where the length of a curve \gamma : [0,1] \rightarrow \mathbb{P}_n is given by

L(\gamma) = \int_0^1 \|\gamma(t)^{-1/2} \gamma'(t) \gamma(t)^{-1/2}\|_2 dt,

where \|\cdot\|_2 denotes the Hilbert-Schmidt norm \|A\|_2 = trace(A^*A). The length of the geodesic (i.e., curve of minimal length) connecting two matrices A, B \in \mathbb{P}_n then defines a distance function on \mathbb{P}_n, \delta(A,B).

Now, the connection to the geometric mean is that M_g(A,B) turns out to be equal to the midpoint of the geodesic connecting A and B! That’s neat, but more importantly, this gives an insight how to define the geometric mean of three (or more) positive definite matrices: simply define M_g(A,B,C) to be the unique point X_0 in the manifold \mathbb{P}_n which minimises the quantity

\delta(A,X)^2 + \delta(B,X)^2 + \delta(C,X)^2.

This “geometric” definition of the geometric mean of positive semidefinite matrices turns out to have all the nice properties that a mean should have (the monotonicity was an open problem, but was resolved a few years ago by Lawson and Lim).

This is a really nice mathematical story, but I was especially happy to hear that these noncommutative geometric means have found highly nontrivial (and important!) applications in various areas of engineering.

In various engineering applications, one makes a measurement such that the result of this measurement is some matrix. Since measurements are noisy, a first approximation for obtaining a clean estimate of the true value of the measured matrix, is to repeat the measurement and take the average, or mean of the measurements. In many applications the most successful (in practice) mean turned out to be the geometric mean as described above. Although the problem of generalising the geometric mean to pairs of matrices and then to tuples of matrices was pursued by Bhatia and his colleagues mostly out of mathematical curiosity, it turned out to be very useful in practice.

2. The Riemann hypothesis and a Schauder basis for \ell^2.

I also have to mention Bhaskar Bagchi’s talk, which stimulated me to go and read his paper “On Nyman, Beurling and Baez-Duarte’s Hilbert space reformulation of the Riemann hypothesis“. The main result (which is essentially an elegant reformulation of a quite old result of Nyman and Beurling, see this old note of Beurling)  is as follows. Let H be the weighted \ell^2 space given by all sequence (x_n)_{n=1}^\infty such that

\sum_n \frac{|x_n|^2}{n^2} < \infty.

In H consider the sequence of vectors:

\gamma_2 = (1/2, 0, 1/2, 0, 1/2, 0,\ldots)

\gamma_3= (1/3, 2/3, 0, 1/3, 2/3, 0, 1/3, 2/3, 0,\ldots)

\gamma_4 = (1/4, 2/4, 3/4, 0, 1/4, 2/4, 3/4, 0, \ldots)

\gamma_5 = (1/5, 2/5, 3/5, 4/5, 0, 1/5, \ldots),

etc. Then Bagchi’s main result is

Theorem: The Riemann Hypthesis is true if and only if the sequence \{\gamma_2, \gamma_3, \ldots \} is total in H

This is interesting, though such results can always be interpreted simply as a claim that the necessary and sufficient condition is now provenly hard. Clearly, nobody expects this to open up a fruitful path by which to approach the Riemann hypothesis, but it gives a nice perspective, as Bagchi writes in his paper:

[The theorem] reveals the Riemann hypothesis as a version of the central theme of harmonic analysis: that more or less arbitrary sequences (subject to mild growth restrictions) can be arbitrarily well approximated by superpositions of a class of simple periodic sequences (in this instance, the sequences \gamma_k).


Souvenirs from the Rocky Mountains

I recently returned from the Workshop on Multivariate Operator Theory at Banff International Research Station (BIRS). BIRS is like the MFO (Oberwolfach): a mathematical resort located in the middle of a beautiful landscape, to where mathematicians are invited to attend/give talks, collaborate, interact, catch up with old friends, make new friends, have fun hike, etc.

As usual I am going over the conference material the week after looking for the most interesting things to write about. This time there were two talks that stood out from my perspective, the one by Richard Rochberg (which was interesting to me because it is on a problem that I have been thinking a lot about), and the one by Igor Klep (which was fascinating because it is about a subject I know little about but wish to learn). There were some other very nice talks, but part of the fun is choosing the best; and one can’t go home and start working on all the new ideas one sees.

A very cool feature of BIRS is that now they automatically shoot the talks and put the videos online (in fact the talks are streamed in real time! If you follow this link at the time of any talk you will see the talk; if you follow the link at any other time it is even better, because there is a webcam outside showing you the beautiful surroundings.

I did not give a talk in the workshop, but I prepared one – here are the slides on the workshop website (best to download and view with some viewer so that the talk unfolds as it should). I also wrote a nice “take home” that would be probably (hopefully) what most people would have taken home from my talk if they heard it, if I had given it. The talk would have been about my recent work with Evgenios Kakariadis on operator algebras associated with monomial ideals (some aspects of which I discussed in a previous post), and here is the succinct Summary (which concentrates on other aspects).  Read the rest of this entry »

Souvenirs from Amsterdam

(I am writing a post on hot trends in mathematics in the midst of war, completely ignoring it. This seems like the wrong thing to do, but my urge to write has overcome me. To any reader of this blog: I wish you a peaceful night, wherever you are).

Last week I returned from the yearly “International Workshop on Operator Theory and Applications”, IWOTA 2014 for short (see the previous post for the topic of my own talk, or this link for the slides).

This conference was very broad (and IWOTA always is). One nice thing about broad conferences is that you are able sometimes to identify a growing trend. In this talk I got particularly excited by a series of talks on “noncommutative function theory” or “free analysis”. There was a special session dedicated to this topic, but I was mostly inspired by a semi-plenary talk by Jim Agler, and also by two interesting talks by Joe Ball and Spela Spenko. I also attended nice talks related to this subject by Victor Vinnikov, Dmitry Kalyuhzni-Verbovetskyi, Baruch Solel, Igor Klep and Bill Helton. This topic has attracted the attention of many operator theorists, for its applications as well as for its inherent beauty, and seems to be accelerating in the last several years; I will only try to give a taste of some neat things that are going on, by telling you about Agler’s talk. What I will not be able to do is to convey Agler’s intense and unique mathematical charisma.

Here is the program of the conference, so you can check out other things that were going on there.

Read the rest of this entry »

Souvenirs from the Black Forest

Last week I attended a workshop titled “Hilbert modules and complex geometry” in MFO (Oberwolfach). In this post I wish to tell about some interesting things that I have learned. There were many great talks to choose from. Below is a sample, in short form, with links.

Read the rest of this entry »

Some links and announcements

  • The course “Advanced Analysis” is over. The lecture notes (the part that I prepared) are available here. Comments are very welcome. I hope to teach this course again in the not too far future and complete the lecture notes (add notes on Banach and C*-algebras, spectral theory and Fredholm theory). The homework exercises are available here, at the bottom of the page (the webpage is in Hebrew but the exercises are in English). 
  • In April Ken Davidson will be visiting our department at BGU. On this occasion we will hold a short conference, dates: April 9-10. Here is the conference page. Contact me for more details.
  • There are some interesting discussions going on in Gowers’s Weblog (see “Why I’ve joined the bad guys” and “Why I’ve joined the good guys” and some of the comments), regarding journals, publishing, new ideas, APCs, and so forth. The big news is that Gowers (after he kind of admits that being an editor of Forum of Mathematics makes him one of the bad guys) is now connected to another publishing adventure, that of epijournals, or arxiv overlay journals, which makes him one of the good guys (Just to set things straight: I think Gowers is a good guy). BTW: Gowers makes it clear that the credit for this initiative does not belong to him but to others, see his post.
  • I promised myself to stop writing about this topic, but I guess I am still allowed to put a link to something that I wrote about this in the past. So here is a link to a letter (also other letters) I sent to Letters to the Editor of the Notices. It is a response to this article by Rob Kirby.

Souvenirs from Bangalore

I recently returned from the two week long workshop and conference Recent Advances in Operator Theory and Operator Algebras which took place in ISI Bangalore. As I promised myself before going, I was on the look-out for something new to be excited about and to learn. The event (beautifully organized and run) was made of two parts: a workshop, which was a one week mini-school on several topics (see here for topics) and a one week conference. It was very very broad, and there were several talks (or informal discussions) which I plan to pursue further.

In this post and also perhaps in a future one I will try to work out (for my own benefit, mostly) some details of a small part of the research presented in two of the talks. The first part is the Superproduct Systems which arise in the theory of E_0-semigroups on type II_1 factors (following the talk of R. Srinivasan). The second (which I will not discuss here, but perhpas in the future) is the equivalence between the Baby Corona Theorem and the Full Corona Theorem (following the mini-course given by B. Wick). In neither case will I describe the most important aspect of the work, but something that I felt was urgent for me to learn. 

Read the rest of this entry »