Inclusions of simple $C^{\ast }$-algebras

Abstract

We prove that if a conditional expectation from a simple $C^{\ast }$-algebra onto its $C^{\ast }$-subalgebra satisfies the Pimsner-Popa inequality, there exists a quasi-basis. As an application, we establish the Galois correspondence for outer actions of finite groups (and more generally finite dimensional Kac algebras) on simple $C^{\ast }$-algebras. We also introduce the notion of sectors for stable simple $C^{\ast }$-algebras and purely infinite simple $C^{\ast }$-algebras in the Cuntz standard form, and discuss several applications, including their relationship to $K$-theory.