%0 Journal Article
%T Inclusions of simple $C^*$-algebras
%A Masaki Izumi
%J Mathematics
%D 2001
%I arXiv
%X We prove that if a conditional expectation from a simple $C^*$-algebra onto its $C^*$-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 $C^*$-algebras. We also introduce the notion of sectors for stable simple $C^*$-algebras and purely infinite simple $C^*$-algebras in the Cuntz standard form, and discuss several applications, including their relationship to $K$-theory.
%U http://arxiv.org/abs/math/0106117v1