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Cartan semigroups and twisted groupoid C*-algebras

Tristan Bice, Lisa Orloff Clark, Ying-Fen Lin, Kathryn McCormick

Abstract

We prove that twisted groupoid C*-algebras are characterised, up to isomorphism, by having Cartan semigroups, a natural generalisation of normaliser semigroups of Cartan subalgebras. This extends the classic Kumjian-Renault theory to general twisted étale groupoid C*-algebras, even non-reduced C*-algebras of non-effective groupoids.

Cartan semigroups and twisted groupoid C*-algebras

Abstract

We prove that twisted groupoid C*-algebras are characterised, up to isomorphism, by having Cartan semigroups, a natural generalisation of normaliser semigroups of Cartan subalgebras. This extends the classic Kumjian-Renault theory to general twisted étale groupoid C*-algebras, even non-reduced C*-algebras of non-effective groupoids.
Paper Structure (19 sections, 77 theorems, 220 equations)

This paper contains 19 sections, 77 theorems, 220 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Normed Spaces
  4. Twists
  5. Twisted Groupoid C*-Algebras
  6. Cartan Semigroups
  7. Algebraic Properties
  8. The Restriction Relation
  9. The Expectation
  10. The Domination Relation
  11. The Ultrafilter Groupoid
  12. Source and Range States
  13. Magnitudes
  14. Angles
  15. Equivalences
  16. ...and 4 more sections

Key Result

Theorem 1.1

[cor:it] Let $A$ be a C*-algebra containing a Cartan semigroup $N$ with semi-Cartan subalgebra $B$ generated by the positive elements of $N$ and a stable expectation $E:A\twoheadrightarrow B$ (see def:semiCartan). We then have an isomorphism $\Psi$ from $A$ onto a twisted groupoid C*-algebra $C=\mat

Theorems & Definitions (166)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • ...and 156 more