Tag: new paper

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 function 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, $\sum X_i X_i < I\}$

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.

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 $H$ is a Hilbert function space on a set $X$, and $k$ its reproducing kernel. The Pick interpolation problem is the following: