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Journal of Operator Theory

Volume 87, Issue 1, Winter 2022  pp. 41-81.

Wold decompositions for representations of $C^*$-algebras associated with noncommutative varieties

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

Summary: Given a set $\mathcal Q$ of polynomials in noncommutative indeterminates $Z_1,\ldots, Z_n$ and a regular domain $\mathcal D_p^{m}(\mathcal H)\subset B(\mathcal H)^n$, $m,n\in \mathbb N$, associated with a positive regular polynomial $p\in \mathbb C\langle Z_1,\ldots, Z_n\rangle $, we consider the variety $$ \mathcal V_\mathcal Q(\mathcal H):=\{X=(X_1,\ldots, X_n)\in \mathcal D_p^{m}(\mathcal H): g({X})=0 \text{ for all } g\in \mathcal Q\}. $$ Each variety $\mathcal V_\mathcal Q(\mathcal H)$ admits a {\it universal model} ${B}=(B_1,\ldots, B_n)$. The main goal of the paper is to study the structure of the $*$-representations of the $C^*$-algebra $C^*(\mathcal V_\mathcal Q)$ generated by $B_1,\ldots, B_n$ and the identity.

Keywords: multivariable operator theory, noncommutative varieties, regular domains, $C^*$-algebras, representations, Wold decompositions, Berezin kernels, classification

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