Bundles of Holomorphic Function Algebras on Subvarieties of the Noncommutative Ball

Abstract

We suggest a general construction of continuous Banach bundles of holomorphic function algebras on subvarieties of the closed noncommutative ball. These algebras are of the form \(\mathcal{A}_d/\overline{I_x}\), where \(\mathcal{A}_d\) is the noncommutative disc algebra defined by G. Popescu, and \(\overline{I_x}\) is the closure in \(\mathcal{A}_d\) of a graded ideal \(I_x\) in the algebra of noncommutative polynomials, depending continuously on a point \(x\) of a topological space \(X\). Moreover, we construct bundles of Fréchet algebras \(\mathcal{F}_d/\overline{I_x}\) of holomorphic functions on subvarieties of the open noncommutative ball. The algebra \(\mathcal{F}_d\) of free holomorphic functions on the unit ball was also introduced by G. Popescu, and \(\overline{I_x}\) stands for the closure in \(\mathcal{F}_d\) of a graded ideal \(I_x\) in the algebra of noncommutative polynomials, depending continuously on a point \(x\in X\).