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

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 $X$.