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$C^*$-algebras associated to directed graphs of groups, and models of Kirchberg algebras

Victor Wu

Abstract

We introduce $C^*$-algebras associated to directed graphs of groups. In particular, we associate a combinatorial $C^*$-algebra to each row-finite directed graph of groups with no sources, and show that this $C^*$-algebra is Morita equivalent to the crossed product coming from the corresponding group action on the boundary of a directed tree. Finally, we show that these $C^*$-algebras (and their Morita equivalent crossed products) contain the class of stable UCT Kirchberg algebras.

$C^*$-algebras associated to directed graphs of groups, and models of Kirchberg algebras

Abstract

We introduce -algebras associated to directed graphs of groups. In particular, we associate a combinatorial -algebra to each row-finite directed graph of groups with no sources, and show that this -algebra is Morita equivalent to the crossed product coming from the corresponding group action on the boundary of a directed tree. Finally, we show that these -algebras (and their Morita equivalent crossed products) contain the class of stable UCT Kirchberg algebras.
Paper Structure (17 sections, 31 theorems, 48 equations)

This paper contains 17 sections, 31 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Graphs and trees
  4. Left cancellative small categories and their $C^*$-algebras
  5. Groupoids of dynamical origin
  6. Graphs of groups
  7. Directed graphs of groups
  8. Directed graphs of groups
  9. Word category
  10. The directed forest $W_{\mathcal{G}_+}$
  11. Directed graph of groups $C^*$-algebras
  12. A $C^*$-algebraic 'directed' Bass--Serre theorem
  13. Groupoid model for $C^*(\mathcal{G}_+)$
  14. Groupoid equivalence
  15. A construction of all stable UCT Kirchberg algebras
  16. ...and 2 more sections

Key Result

Theorem A

Let $\mathcal{G}_+ = (\Gamma_+, G)$ be a countable, row-finite (connected) directed graph of groups with no sources, and choose a base vertex $x \in \Gamma^0$. Write $\pi_1(\mathcal{G}, x)$ for the fundamental group of $\mathcal{G}$ based at $x$, and write $X_{\mathcal{G}_+, x}$ for the directed Bas

Theorems & Definitions (94)

  • Theorem A: Theorem \ref{['thm:morita equiv']}
  • Theorem B: Theorem \ref{['thm:dgog Kirchberg']}
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7: LCSC
  • Remark 8
  • ...and 84 more