Weighted Cuntz Algebras

TL;DR

This paper investigates a class of weighted Cuntz algebras generated by weighted shifts on Fock space, establishing their structure as Cuntz-Pimsner algebras, identifying simplicity conditions, and describing their representations.

Contribution

It introduces a generalized framework for weighted Cuntz algebras, linking them to Cuntz-Pimsner algebras and analyzing their simplicity and representations.

Findings

 algebra is isomorphic to a Cuntz-Pimsner algebra.

 conditions for simplicity of the algebra.

 explicit descriptions of  algebra's representations.

Abstract

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 non simple algebras of this type. We also describe the $C^*$-representations of $\mathcal{T}/\mathcal{K}$.