Weak-* and completely isometric structure of noncommutative function algebras

TL;DR

This paper classifies noncommutative function algebras based on their isometric and weak-* isomorphic properties, linking algebraic isomorphisms to biholomorphic maps between varieties within operator balls.

Contribution

It establishes a classification theorem connecting algebraic isomorphisms with geometric biholomorphisms in noncommutative settings, extending known results.

Findings

Two algebras are isomorphic iff there is a nc biholomorphism between their varieties.

For homogeneous varieties, a linear isomorphism of the operator balls suffices.

Homogeneity is essential; dropping it invalidates the classification.

Abstract

We study operator algebraic and function theoretic aspects of algebras of bounded nc functions on subvarieties of the nc domain determined by all levels of the unit ball of an operator space (nc operator balls). Our main result is the following classification theorem: under very mild assumptions on the varieties, two such algebras $H^\infty(\mathfrak{V})$ and $H^\infty(\mathfrak{W})$ are completely isometrically and weak-* isomorphic if and only if there is a nc biholomorphism between the varieties. For matrix spanning homogeneous varieties in injective operator balls, we further sharpen this equivalence, showing that there exists a linear isomorphism between the respective balls that maps one variety onto the other; in general, we show, the homogeneity condition cannot be dropped. We highlight some difficulties and open problems, contrasting with the well studied case of row ball.