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 $X$.