Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Roberto Hernández Palomares, Miho Mukohara

TL;DR

The paper addresses proper outerness for finite-index endomorphisms and bimodules of simple C*-algebras, extending Izumi's purely infinite results to the general simple case. It proves that any finite-index outer endomorphism is automatically properly outer, and deduces freeness of outer unitary tensor category actions from outerness. As applications, it derives structural properties for irreducible C*-discrete inclusions, including simplicity of intermediate algebras and the BEK property in finite-index situations. The results enhance the classification toolkit for generalized C*-dynamical systems and provide a foundation for analyzing crossed products arising from UTC actions on simple C*-algebras.

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.

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

TL;DR

The paper addresses proper outerness for finite-index endomorphisms and bimodules of simple C*-algebras, extending Izumi's purely infinite results to the general simple case. It proves that any finite-index outer endomorphism is automatically properly outer, and deduces freeness of outer unitary tensor category actions from outerness. As applications, it derives structural properties for irreducible C*-discrete inclusions, including simplicity of intermediate algebras and the BEK property in finite-index situations. The results enhance the classification toolkit for generalized C*-dynamical systems and provide a foundation for analyzing crossed products arising from UTC actions on simple C*-algebras.

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.

Paper Structure

This paper contains 7 sections, 25 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Properly outer endomorphisms
  4. Proof of Theorem \ref{['thmalpha:end']}
  5. Applications
  6. Freeness of UTC actions
  7. Appendix Section

Key Result

Theorem 1

Let $A$ be a separable simple C$^\ast$-algebra and $\rho\colon A\to A$ be a finite index outer endomorphism (i.e., $\{T\in M(A)|\ \rho(a)T = Ta\ \forall a\in A\}= 0$). Then, $\rho$ is properly outer.

Theorems & Definitions (72)

  • Theorem 1: Theorem \ref{['thm end']}
  • Corollary 2: Corollary \ref{['cor free']}
  • Corollary 3: Corollary \ref{['cor csirr']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6: Theorem 2.8 of MR1900138
  • Theorem 2.7: Corollary 3.7 of MR1900138
  • ...and 62 more