Noncommutative Analysis

Month: February, 2017

Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball

Guy Salomon, Eli Shamovich and I recently uploaded to the arxiv our paper “Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball“. This paper blends in with the current growing interest in noncommutative functions theory, continues and unifies several strands of my past research.

A couple of years ago, after being inspired by lectures of Agler, Ball, McCarthy and  Vinnikov on the subject, and after years of being influenced by Paul Muhly and Baruch Solel’s work, I realized that many of my different research projects (subproduct systems, the isomorphism problem, space of Dirichlet series with the complete Pick property, operator algebras associated with monomial ideals) are connected by the unifying theme of bounded analytic nc functions on subvarieties of the nc ball. “Realized” is a strong word, because many of my original ideas on this turned out to be false, and others I still don’t know how to prove. Anyway, it took me a couple of years and a lot of help, and here is this paper.

In short, we study algebras of bounded analytic functions on subvarieties of the the noncommutative (nc) unit ball :

\mathfrak{B}_d = \{(X_1, \ldots, X_d) tuples of n \times n matrices,  \sum X_i X_i \leq I\}

as well as bounded analytic functions that extend continuously to the “boundary”. We show that these algebras are multiplier algebras of appropriate nc reproducing kernel Hilbert spaces, and are completely isometrically isomorphic to the quotient of H^\infty(\mathfrak{B}_d) (the bounded nc analytic functions in the ball) by the ideal of nc functions vanishing on the variety. We classify these algebras in terms of the varieties, similar to classification results in the commutative case. We also identify previously studied algebras (such as multiplier algebras of complete Pick spaces and tensor algebras of subproduct systems) as algebras of bounded analytic functions on nc varieties. See the introduction for more.

We certainly plan to continue this line of research in the near future – in particular, the passage to other domains (beyond the ball), and the study of algebraic/bounded isomorphisms.

A First Course in Functional Analysis (my book)

She’hechiyanu Ve’kiyemanu!

My book, A First Course in Functional Analysis, to be published with Chapman and Hall/CRC, will soon be out. There is already a cover, check it out on the CRC Press website.

This book is written to accompany an undergraduate course in functional analysis, where the course I had in mind is precisely the course that we give here at the Technion, with the same constraints. Constraint number 1: a course in measure theory is not mandatory in our undergraduate program. So how can one seriously teach functional analysis with significant applications? Well, one can, and I hope that this book proves that one can. I already wrote before, measure theory is not a must. Of course anyone going for a graduate degree in math should study measure theory (and get an A), but I’d like the students to be able to study functional analysis before that (so that they can do a masters degree in operator theory with me).

I believe that the readers will find many other original organizational contributions to the presentation of functional analysis in this book, but I leave them for you to discover. Instructors can request an e-copy for inspection (in the link to the publisher website above), friends and direct students can get a copy from me, and I hope that the rest of the world will recommend this book to their library (or wait for the libgen version).