Journal of Operator Theory

Volume 87, Issue 1, Winter 2022 pp. 83-111.

Weighted Cuntz algebras

Authors: Leonid Helmer (1), Baruch Solel (2)
Author institution:(1) Department of Mathematics, Ben Gurion University, Beer Sheva, Israel
(2) Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel

Summary: We study the $C^*$ -algebra $\mathcal{T}/\mathcal{K}$ where $\mathcal{T}$ is the $C^*$ -algebra generated by $d$ weighted shifts on the Fock space of $\mathbb{C}^d$ , $\mathcal{F}(\mathbb{C}^d)$ , (where the weights are given by a sequence $\{Z_k\}$ of matrices $Z_k\in M_{d^k}(\mathbb{C})$ ) and $\mathcal{K}$ is the algebra of compact operators on the Fock space. If $Z_k=I$ for every $k$ , $\mathcal{T}/\mathcal{K}$ is the Cuntz algebra $\mathcal{O}_d$ . We show that $\mathcal{T}/\mathcal{K}$ is isomorphic to a Cuntz--Pimsner algebra and use it to find conditions for the algebra to be simple. We present examples of simple and of nonsimple algebras of this type. We also describe the $C^*$ -representations of $\mathcal{T}/\mathcal{K}$ .

DOI: http://dx.doi.org/10.7900/jot.2020jul02.2313
Keywords: weighted shift, simplicity, Cuntz--Pimsner algebra, Fock space, $C^*$ -correspondence, $C^*$ -algebra

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