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

Robert Valente, Dilian Yang

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.

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

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 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.
Paper Structure (22 sections, 32 theorems, 82 equations)

This paper contains 22 sections, 32 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Single-vertex rank $\mathsf k$ graphs
  4. Self-similar single-vertex $\mathsf k$-graph C*-algebras
  5. Right LCM semigroup C*-algebras and their boundary quotient C*-algebras
  6. Semigroups from self-similar actions
  7. Rank 1 case: more than GBS semigroups
  8. Some basic properties
  9. Pseudo-freeness of $({\mathbb{Z}}, \mathsf{E}_n)$
  10. The periodicity of $({\mathbb{Z}}, \mathsf{E}_n)$
  11. The case of $\kappa=1$
  12. The general case
  13. Cartan subalgebras of ${\mathcal{O}}_{{\mathbb{Z}}, \mathsf{E}_n}$
  14. The case of $n=1$
  15. The case of $n>1$
  16. ...and 7 more sections

Key Result

Proposition 2.7

Let $(G,\Lambda_\theta)$ be a self-similar $\mathsf k$-graph. Then

Theorems & Definitions (83)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4: LY19$_2$
  • Definition 2.5
  • Definition 2.6: LY19$_2$
  • Proposition 2.7
  • Theorem 2.8: Li-Yang LY19$_2$
  • Definition 2.9
  • Remark 2.10
  • ...and 73 more