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

Abstract

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