Irreducible inclusions of simple C$^*$-algebras

TL;DR

This paper systematically studies C$^*$-irreducible inclusions of simple C$^*$-algebras, providing intrinsic characterizations, examples from groups and dynamical systems, and analyzing their behavior under various constructions.

Contribution

It offers a new intrinsic characterization of C$^*$-irreducible inclusions and explores their properties and examples from groups, dynamical systems, and inductive limits.

Findings

Characterization of C$^*$-irreducibility in terms of algebraic properties

Construction methods for C$^*$-irreducible inclusions from groups and dynamical systems

Behavior of C$^*$-irreducibility under tensor products and inductive limits

Abstract

The literature contains interesting examples of inclusions of simple C$^*$-algebras with the property that all intermediate C$^*$-algebras likewise are simple. In this article we take up a systematic study of such inclusions, which we refer to as being C$^*$-irreducible by the analogy that all intermediate von Neumann algebras of an inclusion of factors are again factors precisely when the given inclusion is irreducible. We give an intrinsic characterization of when an inclusion of C$^*$-algebras is C$^*$-irreducible, and use this to revisit known and exhibit new C$^*$-irreducible inclusions arising from groups and dynamical systems. Using a theorem of Popa one can show that an inclusion of II$_1$-factors is C$^*$-irreducible if and only if it is irreducible with finite Jones index. We further show how one can construct C$^*$-irreducible inclusions from inductive limits, and we discuss how the notion of C$^*$-irreducibility behaves under tensor products.