Some Results on Inner Quasidiagonal $C^*$-algebras

TL;DR

This paper investigates properties of inner quasidiagonal $C^*$-algebras, proving stability under certain operations, characterizing quasidiagonality in just-infinite cases, and computing entropy dimensions.

Contribution

It establishes that crossed products by Rokhlin actions preserve strong quasidiagonality and characterizes quasidiagonality in just-infinite $C^*$-algebras, also computing their topological free entropy dimension.

Findings

Crossed product by Rokhlin action remains strongly quasidiagonal.

A just-infinite $C^*$-algebra is quasidiagonal iff it is inner quasidiagonal.

Topological free entropy dimension computed for just-infinite $C^*$-algebras.

Abstract

In the current article, we prove the cross product $C^*$-algebra by a Rokhlin action of finite group on a strongly quasidiagonal $C^*$-algbra is strongly quasidiagonal again. We also show that a just-infinite $C^*$-algebra is quasidiagonal if and only if it is inner quasidiagonal. Finally, we compute the topological free entropy dimension in just-infinite $C^*$-algebras.