Decomposable approximations and approximately finite dimensional C*-algebras

TL;DR

This paper proves that nuclear C*-algebras with certain convex combination approximations are approximately finite dimensional, advancing understanding of their structure and approximation properties.

Contribution

It establishes a new link between completely positive approximations with order zero summands and the AF property in nuclear C*-algebras.

Findings

Nuclear C*-algebras with convex combination approximations are AF.

Approximate finite dimensionality follows from specific approximation conditions.

Provides new criteria for AF-ness in nuclear C*-algebras.

Abstract

Nuclear $C^*$-algebras having a system of completely positive approximations formed with convex combinations of a uniformly bounded number of order zero summands are shown to be approximately finite dimensional.