Properly Outer Actions of Tensor Categories on C$^*$-algebras

TL;DR

This paper proves that proper outerness automatically holds for finite index outer endomorphisms of simple C$^*$-algebras, ensuring freeness of outer tensor category actions and deriving structural results for C$^*$-algebra inclusions.

Contribution

It extends the concept of proper outerness to finite index endomorphisms and bimodules, establishing automatic proper outerness and freeness in this setting.

Findings

Proper outerness holds automatically for finite index outer endomorphisms.

Freeness of outer actions of unitary tensor categories on simple C$^*$-algebras is automatic.

Structural results about C$^*$-irreducibility of algebra inclusions are obtained.

Abstract

We discuss proper outerness for finite index endomorphisms and finite index bimodules of simple C$^*$-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$^*$-algebras. Consequently, freeness for outer actions of unitary tensor categories on simple C$^*$-algebras is also shown to hold automatically. As applications, we obtain structural results about potentially infinite index irreducible discrete inclusions of C$^*$-algebras, such as C$^*$-irreducibility.