Semigroups of self-similar actions and higher rank Baumslag-Solitar semigroups

TL;DR

This paper explores higher rank Baumslag-Solitar semigroups, analyzing their associated C*-algebras, von Neumann algebra types, and Cartan subalgebras, using self-similar higher rank graph techniques.

Contribution

It introduces the study of higher rank Baumslag-Solitar semigroups, characterizes von Neumann algebra factoriality and types, and constructs canonical Cartan subalgebras for these semigroups.

Findings

Characterized factoriality and types of von Neumann algebras for certain classes.

Determined canonical Cartan subalgebras for Furstenberg-related semigroups.

Extended analysis to generalized Baumslag-Solitar semigroups in rank 1.

Abstract

In this paper, we initiate the study of higher rank Baumslag-Solitar semigroups and their related C*-algebras. We focus on two extreme, but interesting, classes - one is related to products of odometers and the other is related to Furstenberg's $\times p,, \times q$ conjecture. For the former class, whose C*-algebras are studied by H. Li and the second author, we here characterize the factoriality of the associated von Neumann algebras and further determine their types; for the latter, we obtain their canonical Cartan subalgebras. In the rank 1 case, we study a more general setting which encompasses (single-vertex) generalized Baumslag-Solitar semigroups. One of our main tools is from self-similar higher rank graphs and their C*-algebras.