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Journal of Operator Theory

Volume 90, Issue 1,  Summer 2023  pp. 91-170.

Noncommutative varieties, universal operator models, and operator algebras

Authors:  Gelu Popescu
Author institution: Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, U.S.A.

Summary:  The goal of this paper is to introduce large classes of noncommutative varieties in non-regular noncommutative domains in $B(\mathcal H)^n$, where $B(\mathcal H)$ is the algebra of all bounded linear operators on a Hilbert space $\mathcal H$, and study them in connection with their universal models and the Hardy algebras and $C^*$-algebras they generate. The multivariable operator theory of these varieties is developed throughout the paper. This includes, in particular, the study of non-regular commutative domains generated by admissible free holomorphic functions and interpolation for the multipliers of the weighted symmetric Fock spaces and the corresponding reproducing kernel Hilbert spaces on domains in $\mathbb C^n$.

Keywords:  Multivariable operator theory, noncommutative varieties, universal operator models, Fock spaces, noncommutative Hardy algebras, $C^*$-algebras, multipliers, commutant lifting, interpolation.

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