On simplicity of Cuntz algebras and its applications

Cuntz algebra O2 is the universal C*-algebra generated by two isometries s1, s2 satisfying s1s1*+s2s2*=1. This is separable, simple, infinite C*-algebra containing a copy of any nuclear C*-algebra. The C*-algebra O2 plays a central role in the modern theory of C*-algebras and appears in many fundamental statements, including a formulation of the celebrated Uniform Coefficient Theorem (UCT). There are several extensions of this notion, including Cuntz algebra On, Cuntz-Krieger algebra FA for a matrix A, Cuntz-Pimsner algebra OX and its relaxation by Katsura for a C*-correspondence X, and Cuntz-Nica-Pimsner algebra NOX, for a product system X. We give an overview of the construction of these classes of C*-algebras with a focus on conditions ensuring their simplicity, which is needed in the Elliott Classification Program, as it stands now. The results we present are now part of the literature, but we hope to shed a light on recent developments in a fascinating area of modern operator algebras.