A Note On Inner Quasidiagonal C*-Algebras

TL;DR

This paper introduces two new ways to characterize separable inner quasidiagonal C*-algebras and demonstrates that certain free product constructions preserve this property, with applications to AF algebras.

Contribution

It provides novel characterizations of separable inner quasidiagonal C*-algebras and proves their stability under specific free product operations.

Findings

Full free products of two inner quasidiagonal C*-algebras are inner quasidiagonal.

Free products with amalgamation over a full matrix algebra preserve inner quasidiagonality.

Under certain conditions, free products of AF algebras with amalgamation are inner quasidiagonal.

Abstract

In the paper, we give two new characterizations of separable inner quasidiagonal C*-algebras. Base on these characterizations, we show that a unital full free product of two inner quasidiagonal C*-algebras is inner quasidiagonal again. As an application, we show that a unital full free product of two inner quasidiagoanl C*-algebras with amalgmation over a full matrix algebra is inner quasidiagonal. Meanwhile, we conclude that a unital full free product of two AF algebras with amalgamation over a finite-dimensional C*-algebra is inner quasidiagonal if there are faithful tracial states on each of these two AF algebras such that the restrictions on the common subalgebra agree.