Weak-* and completely isometric structure of noncommutative function algebras
Jeet Sampat, Orr Shalit
TL;DR
The paper tackles the problem of classifying algebras H^\infty(\mathfrak{V}) of bounded nc functions on nc subvarieties of operator balls by geometric data. It develops a canonical weak-* dual framework and proves that, under mild hypotheses, weak-* completely isometric isomorphisms correspond to nc biholomorphisms between the varieties, with a stronger linear-ball mapping result for matrix-spanning homogeneous subvarieties in injective balls. It also identifies fundamental obstacles when moving beyond homogeneous or injective settings, notably via boundary-value phenomena and fibered-representation analysis, and provides partial positive results under regularity assumptions. The work bridges nc function theory and operator-algebraic classification, clarifying when algebraic isomorphisms arise from geometric isomorphisms and outlining concrete open problems related to representations and extensions in general nc operator balls.
Abstract
We study operator algebraic and function theoretic aspects of algebras of bounded nc functions on subvarieties of the nc domain determined by all levels of the unit ball of an operator space (nc operator balls). Our main result is the following classification theorem: under very mild assumptions on the varieties, two such algebras $H^\infty(\mathfrak{V})$ and $H^\infty(\mathfrak{W})$ are completely isometrically and weak-* isomorphic if and only if there is a nc biholomorphism between the varieties. For matrix spanning homogeneous varieties in injective operator balls, we further sharpen this equivalence, showing that there exists a linear isomorphism between the respective balls that maps one variety onto the other; in general, we show, the homogeneity condition cannot be dropped. We highlight some difficulties and open problems, contrasting with the well studied case of row ball.