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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 $\mathcal{T}/\mathcal{K}$.

Keywords: weighted shift, simplicity, Cuntz--Pimsner algebra, Fock space, $C^*$-correspondence, $C^*$-algebra

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