Journal of Operator Theory

Volume 93, Issue 1, Winter 2025 pp. 37-89.

Noncommutative domains, 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: Let $B(\mathcal H)$ be the algebra of all bounded linear operators on a Hilbert space $\mathcal H$. The main goal of the paper is to find large classes of noncommutative domains in $B(\mathcal H)^n$ with prescribed universal operator models, acting on the full Fock space with $n$ generators, and to study these domains and their universal models in connection with the Hardy algebras and the $C^*$-algebras they generate. While the class of these domains contains the regular noncommutative domains previously studied in the literature, the main focus of the present paper is on the non-regular domains. The multi-variable operator theory of these domains is developed throughout the paper.

DOI: http://dx.doi.org/10.7900/jot.2022nov17.2407
Keywords: multivariable operator theory, noncommutative domains, universal operator models, Fock spaces, noncommutative Hardy algebras, $C^*$-algebra

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