$C^*$-simplicity, confined subalgebras, and operator algebraic uniform recurrence

TL;DR

This paper introduces confined subalgebras and Uniformly Recurrent States in operator algebras, providing a new characterization of $C^*$-simplicity for countable discrete groups.

Contribution

It generalizes Kennedy's characterization of $C^*$-simplicity by linking it to the absence of non-trivial amenable confined subalgebras.

Findings

A countable discrete group is $C^*$-simple iff it has no non-trivial amenable confined subalgebras.

Introduces the concepts of confined subalgebras and Uniformly Recurrent States.

Extends Kennedy's result to a broader operator algebraic framework.

Abstract

We introduce the notion of confined subalgebras in the context of the group von Neumann algebra. We also define Uniformly Recurrent States -- an operator-algebraic analog of Uniformly Recurrent Subgroups. Using this framework, we show that a countable discrete group is $C^*$-simple if and only if it admits no non-trivial amenable confined subalgebras. This generalizes the well-known result of Kennedy that characterizes $C^*$-simplicity in terms of trivial amenable uniformly recurrent subgroups.