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Cartan subalgebras of C*-algebras associated with iterated function systems

Kei Ito

Abstract

Let $X$ be a compact Hausdorff space, and let $γ$ be an iterated function system on $X$. Kajiwara and Watatani showed that if $γ$ is self-similar and satisfies the open set condition and some additional technical conditions, $C(X)$ is a maximal abelian subalgebra of the Kajiwara--Watatani algebra $Ø_γ$ associated with $γ$. In this paper, we extend their results by providing sufficient conditions under which $C(X)$ becomes a Cartan subalgebra of $Ø_γ$. Additionally, we present sufficient conditions under which $C(X)$ fails to be a masa of $Ø_γ$.

Cartan subalgebras of C*-algebras associated with iterated function systems

Abstract

Let be a compact Hausdorff space, and let be an iterated function system on . Kajiwara and Watatani showed that if is self-similar and satisfies the open set condition and some additional technical conditions, is a maximal abelian subalgebra of the Kajiwara--Watatani algebra associated with . In this paper, we extend their results by providing sufficient conditions under which becomes a Cartan subalgebra of . Additionally, we present sufficient conditions under which fails to be a masa of .
Paper Structure (28 sections, 78 theorems, 160 equations)

This paper contains 28 sections, 78 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Historical background
  3. Main theorems
  4. Structure of the Paper
  5. Preparations for Operator Algebras
  6. Adjointable operators between right Hilbert modules
  7. $X$-squared matrices
  8. Cuntz--Pimsner--Katsura algebras
  9. The group C*-algebra $C^*({\mathbb Z})$
  10. Preparations for Iterated Function Systems
  11. Definitions and Fundamental Properties
  12. The binary operator $\boxdot$
  13. The binary operator $\boxtimes$
  14. Review of Kajiwara–Watatani Algebras
  15. Certain Representations of Kajiwara--Watatani Correspondences
  16. ...and 13 more sections

Key Result

Theorem 1.2.1

If $\gamma$ is not essentially free, then $\hat{\rho}_A(A)$ is properly contained in $\hat{D}$ and hence is not a masa of $C^*(\hat{\rho})$.

Theorems & Definitions (190)

  • Theorem 1.2.1: Theorem \ref{['5.5.5']}
  • Theorem 1.2.2: Theorem \ref{['5.5.5']} and Theorem \ref{['5.5.7']}
  • Theorem 1.2.3: Theorem \ref{['5.9.8']} and Theorem \ref{['5.10.2']}
  • Definition 2.2.1
  • Definition 2.2.2
  • Definition 2.2.3
  • Proposition 2.3.2
  • proof
  • Definition 2.3.4: K, Definition 5.6
  • Proposition 2.4.2
  • ...and 180 more