C*-algebras whose every C*-subalgebra is AF

TL;DR

This paper characterizes scattered C*-algebras as those whose all subalgebras are approximately finite-dimensional (AF) and have real rank zero, establishing equivalences among these properties.

Contribution

It proves the equivalence between scatteredness, all subalgebras being AF, and all subalgebras having real rank zero in C*-algebras.

Findings

Scattered C*-algebras have all subalgebras AF.

All subalgebras of a scattered C*-algebra have real rank zero.

The paper establishes the equivalence of these properties.

Abstract

Let $A$ be a $C^*$-algebra. It is shown that the following conditions are equinvalent: (1) $A$ is scattered, (2) every $C^*$-subalgebra of $A$ is AF, (3) every $C^*$-subalgebra of $A$ has real rank zero.