Continuous deformations of algebras of holomorphic functions on subvarieties of the noncommutative ball

The talk is devoted to continuous deformations of algebras of holomorphic functions on homogeneous subvarieties of the noncommutative ball (either open or closed). Such algebras can be defined as $A/I$ and $F/I$, where $A$ and $F$ are the algebras of free holomorphic functions on the closed (respectively, open) ball introduced by G. Popescu, and where the ideal $I$ determines a noncommutative subvariety. Alternatively, these algebras can be defined in the framework of the “matricial” theory of free noncommutative functions, which goes back to J. Taylor and which was subsequently developed by Vinnikov, Kaliuzhnyi-Verbovetskyi, Agler, McCarthy and others. The definitions of all these objects will be given in our talk.
To give a precise meaning to the notion of a “continuous deformation”, we recall the definitions of Banach and locally convex algebra bundles, and we construct bundles with fibers isomorphic to $A/I_x$ and $F/I_x$, respectively, where the ideal $I_x$ depends continuously (in a suitable sense) on a point $x$ in a topological space $X$.