Properly Outer Actions of Tensor Categories on C${}^{\ast }$-algebras

Abstract

We discuss proper outerness for finite index endomorphisms and finite index bimodules of simple C${}^{\ast }$-algebras, extending recent similar results by Izumi concerning the purely infinite setting. Our main result is that proper outerness holds automatically for finite index outer endomorphisms of simple C${}^{\ast }$-algebras. Consequently, freeness for outer actions of unitary tensor categories on simple C${}^{\ast }$-algebras is also shown to hold automatically. As applications, we obtain structural results about potentially infinite index irreducible discrete inclusions of C${}^{\ast }$-algebras, such as C${}^{\ast }$-irreducibility.

36 pages