On simplicity of Cuntz algebra and its generalizations

Abstract

Cuntz algebra ${\mathcal{O}}_2$ is the universal $C^{\ast }$-algebra generated by two isometries $s_1,s_2$ satisfying $s_1{s}_1^{\ast }+s_2{s}_2^{\ast }=1$. This is separable, simple, infinite $C^{\ast }$-algebra containing a copy of any nuclear $C^{\ast }$-algebra. The $C^{\ast }$-algebra ${\mathcal{O}}_2$ plays a central role in the modern theory of $C^{\ast }$-algebras and appears in many substantial statements, including a formulation of the celebrated Uniform Coefficient Theorem (UCT). There are several extensions of this notion, including Cuntz algebra ${\mathcal{O}}_n$, Cuntz-Krieger algebra ${\mathcal{O}}_A$ for a matrix $A$, Cuntz-Pimsner algebra ${\mathcal{O}}_X$ and its relaxation by Katsura for a $C^{\ast }$-correspondence $X$, and Cuntz-Nica-Pimsner algebra ${\mathcal{N}\mathcal{O}}_X$, for a product system $X$. We give an overview of the construction of these classes of $C^{\ast }$-algebras with a focus on conditions ensuring their simplicity, which is needed in the Elliott Classification Program, as it erature, except our discussion on the sufficient conditions for simplicity of the reduced Cuntz-Nica-Pimsner algebra ${\mathcal{N}\mathcal{O}}_X^r$, which is known to expertsstands now. The results we present are now part of the lit, but might happen to be new for some of our audiences.

This is a survey