Continuous Deformations of Algebras of Holomorphic Functions on Subvarieties of a Noncommutative Ball

TL;DR

This paper develops a method to construct continuous bundles of algebras of holomorphic functions on subvarieties of noncommutative balls, generalizing classical function algebra theory to a noncommutative setting.

Contribution

It introduces a framework for continuous Banach bundles of noncommutative holomorphic function algebras parametrized by a topological space.

Findings

Constructed bundles with fibers as quotients of noncommutative disk algebra

Constructed bundles with fibers as quotients of free holomorphic function algebra

Established continuous dependence of ideals on base points

Abstract

We propose a general method for constructing continuous Banach bundles whose fibers are algebras of holomorphic functions on subvarieties of a closed noncommutative ball. These algebras are of the form $\mathcal{A}_d/I_x$, where $\mathcal{A}_d$ is the noncommutative disk algebra introduced by G. Popescu, and $I_x$ is a graded ideal in $\mathcal{ A}_d$, which depends continuously on the point $x$ of the topological space $X$. Similarly, we construct bundles with fibers isomorphic to the algebras $\mathcal{F}_d/I_x$ of holomorphic functions on subvarieties of an open noncommutative ball. Here $\mathcal{F}_d$ is the algebra of free holomorphic functions on the unit ball, which was also introduced by G. Popescu, and $I_x$ is a graded ideal in $\mathcal{F}_d$, which depends continuously on the point $x$ of the topological space $X$.