$C^{\ast }$-simplicity, confined subalgebras, and operator algebraic uniform recurrence

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 $C^{\ast }$-simple if and only if it admits no non-trivial amenable confined subalgebras. This generalizes the well-known result of Kennedy that characterizes $C^{\ast }$-simplicity in terms of trivial amenable uniformly recurrent subgroups.

24 pages; comments are welcome