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 $x\in X$.
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