INCLUSIONS OF SIMPLE C^{∗}-ALGEBRAS

MASAKI IZUMI

Abstract. We prove that if a conditional expectation from a simple C^{∗}-algebra onto its C^{∗}-subalgebra satisfies the Pimsner-Popa inequality, there exists a quasibasis. 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.