On the classification of function algebras on subvarieties of noncommutative operator balls

TL;DR

This paper investigates the structure of algebras of bounded noncommutative functions on operator space balls and their subvarieties, revealing that these algebras encode the geometry of the underlying varieties and operator spaces.

Contribution

It establishes that the algebra of uniformly continuous nc functions on homogeneous subvarieties uniquely determines the nc variety up to biholomorphism and operator space isomorphism.

Findings

The nc algebra may not be the multiplier algebra of any nc RKHS.

The nc variety is a complete invariant for the algebra of functions on homogeneous subvarieties.

Biholomorphisms between varieties extend to ambient nc operator balls.

Abstract

We study algebras of bounded noncommutative (nc) functions on unit balls of operator spaces (nc operator balls) and on their subvarieties. Considering the example of the nc unit polydisk we show that these algebras, while having a natural operator algebra structure, might not be the multiplier algebra of any reasonable nc reproducing kernel Hilbert space (RKHS). After examining additional subtleties of the nc RKHS approach, we turn to study the structure and representation theory of these algebras using function theoretic and operator algebraic tools. We show that the underlying nc variety is a complete invariant for the algebra of uniformly continuous nc functions on a homogeneous subvariety, in the sense that two such algebras are completely isometrically isomorphic if and only if the subvarieties are nc biholomorphic. We obtain extension and rigidity results for nc maps between subvarieties of nc operator balls corresponding to injective spaces that imply that a biholomorphism between homogeneous varieties extends to a biholomorphism between the ambient balls, which can be modified to a linear isomorphism. Thus, the algebra of uniformly continuous nc functions on nc operator balls, and even its restriction to certain subvarieties, completely determine the operator space up to completely isometric isomorphism.