Noncommutative Analysis

Month: January, 2017

Introduction to von Neumann algebras (Topics in functional analysis 106433 – Spring 2017)

This coming spring semester, I will be giving a graduate course, “Introduction to von Neumann algebras”. This will be a rather basic course, since most of our graduate students haven’t had much operator algebras. (Unfortunately, most of our graduate students didn’t all take the topics course I gave the previous spring). In any sub-field of operator theory, operator algebras, and noncommutative analysis, von Neumann algebras appear and are needed. Thus, this course is meant first and foremost to give (prospective) students and postdocs in our group the opportunity to add this subject to the foundational part of their training. This course is also an opportunity for me to refurbish and reorganize the working knowledge that I acquired during several years of occasional encounters with this theory. Finally, I believe that this course could be really interesting to other serious students of mathematics, who will have many occasions to bump into von Neumann algebras, regardless of the specific research topic that they decide to devote themselves to (yes, you too!).

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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, and, as usual, writing this post is meant mostly as a diversion for me (somewhere between doing “real” work and getting frustrated about the news), just giving some background and stating 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 H is a Hilbert function space on a set X, and k its reproducing kernel. The Pick interpolation problem is the following:

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