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C*-diagonals with Cantor spectrum in Cuntz algebras

Samuel Evington, Philipp Sibbel

Abstract

We prove that there exists a C*-diagonal with Cantor spectrum in the Cuntz algebra $\mathcal{O}_k$ for $2 \le k < \infty$. Our method generalises to an uncountable family of UCT Kirchberg algebras with distinct K-theory. Moreover, we construct principal étale groupoid models for these Cuntz algebras and UCT Kirchberg algebras.

C*-diagonals with Cantor spectrum in Cuntz algebras

Abstract

We prove that there exists a C*-diagonal with Cantor spectrum in the Cuntz algebra for . Our method generalises to an uncountable family of UCT Kirchberg algebras with distinct K-theory. Moreover, we construct principal étale groupoid models for these Cuntz algebras and UCT Kirchberg algebras.

Paper Structure

This paper contains 9 sections, 26 theorems, 43 equations.

Table of Contents

  1. Preliminaries
  2. Notation
  3. Cartan subalgebras and C*-diagonals
  4. Étale Groupoids
  5. K-theoretic preliminaries
  6. The main construction
  7. K-theory of B(k,alpha)
  8. C*-diagonals in Cuntz algebras
  9. Groupoid models

Key Result

Theorem A

The Cuntz algebra $\mathcal{O}_k$ contains a C$^*$-diagonal with Cantor spectrum for every $2 \leq k < \infty$.

Theorems & Definitions (54)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 1: cf. renault2008cartan
  • Theorem 2: renault2008cartan
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • ...and 44 more