A note on unital full amalgamated free products of quasi-diagonal C*-algebras

TL;DR

This paper investigates conditions under which unital full amalgamated free products of quasidiagonal C*-algebras remain quasidiagonal, providing criteria involving finite-dimensional subalgebras and faithful tracial states.

Contribution

It establishes a sufficient condition for the quasidiagonality of amalgamated free products of quasidiagonal C*-algebras over finite-dimensional subalgebras.

Findings

A unital full free product of two AF algebras with compatible faithful tracial states over a finite-dimensional subalgebra is AF.

The paper provides a criterion ensuring the quasidiagonality of amalgamated free products.

It extends understanding of how quasidiagonality behaves under free product constructions.

Abstract

In the paper, we consider the question whether a unital full amalgamated free product of quasidiagonal C*-algebras is quasidiagonal again. We give a sufficient condition such that a unital full amalgamated free product of quasidiagonal C*-algebras with amalgamation over a finite dimensional C*- algebra is quasidiagonal. Applying this result, we conclude that a unital full free product of two AF algebras with amalgamation over a finite-dimensional C*-algebra is AF if there are faithful tracial states on each of these two AF algebras such that the restrictions on the common subalgebra agree.