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
$\mathbb C^n$.
DOI: http://dx.doi.org/10.7900/jot.2021oct12.2377
Keywords: Multivariable operator theory, noncommutative varieties,
universal operator models, Fock spaces, noncommutative Hardy
algebras, $C^*$-algebras, multipliers, commutant lifting, interpolation.
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