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Cartan subalgebras of C*-algebras associated with complex dynamical systems

Kei Ito

TL;DR

The paper addresses when the Cartan subalgebra C0(X) sits inside the KW algebra O_R(X) associated to a complex dynamical system defined by a rational map R. It introduces a universal covariant representation of KW C*-correspondences, realizes O_R(X) as the C*-algebra generated by this representation, and derives a sharp criterion: C0(X) is Cartan in O_R(X) precisely when X has no branch points of R. The approach blends operator-algebraic constructions (Cuntz–Pimsner framework, representations, gauge actions) with the dynamics of R (branch points, Fatou/Julia sets) to link algebraic regularity to dynamical singularities. This provides both a concrete model for O_R(X) and a decisive criterion with potential implications for groupoid twists and Renault–Kumjian theory in complex dynamics.

Abstract

Let $R$ be a rational function with degree $\geq 2$ and $X$ be its Julia set, its Fatou set, or the Riemann sphere. Suppose that $X$ is not empty. We can regard $R$ as a continuous map from $X$ onto itself. Kajiwara and Watatani showed that in the case that $X$ is the Julia set, $C_0(X)$ is a maximal abelian subalgebra of $\mathcal{O}_R(X)$, where $\mathcal{O}_R(X)$ denotes the C*-algebra associated with the dynamical system $(X,R)$ introduced by them. In this paper, we develop their result and give the equivalent condition for $C_0(X)$ to be a Cartan subalgebra of $\mathcal{O}_R(X)$.

Cartan subalgebras of C*-algebras associated with complex dynamical systems

TL;DR

The paper addresses when the Cartan subalgebra C0(X) sits inside the KW algebra O_R(X) associated to a complex dynamical system defined by a rational map R. It introduces a universal covariant representation of KW C*-correspondences, realizes O_R(X) as the C*-algebra generated by this representation, and derives a sharp criterion: C0(X) is Cartan in O_R(X) precisely when X has no branch points of R. The approach blends operator-algebraic constructions (Cuntz–Pimsner framework, representations, gauge actions) with the dynamics of R (branch points, Fatou/Julia sets) to link algebraic regularity to dynamical singularities. This provides both a concrete model for O_R(X) and a decisive criterion with potential implications for groupoid twists and Renault–Kumjian theory in complex dynamics.

Abstract

Let be a rational function with degree and be its Julia set, its Fatou set, or the Riemann sphere. Suppose that is not empty. We can regard as a continuous map from onto itself. Kajiwara and Watatani showed that in the case that is the Julia set, is a maximal abelian subalgebra of , where denotes the C*-algebra associated with the dynamical system introduced by them. In this paper, we develop their result and give the equivalent condition for to be a Cartan subalgebra of .
Paper Structure (24 sections, 60 theorems, 72 equations)

This paper contains 24 sections, 60 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Historical background
  3. Main theorem
  4. Preparations for Operator Algebras
  5. Adjointable operators between right Hilbert modules
  6. $X$-squared matrices
  7. Cuntz--Pimsner algebras
  8. $Z$-squared $B$-valued matrices
  9. The reduced group C*-algebra $C^*({\mathbb Z})$
  10. Review of Complex Dynamical Systems
  11. Branch points of holomorphic functions
  12. Branch points and fixed points of rational functions
  13. Fatou and Julia sets
  14. Review of Kajiwara--Watatani Algebras
  15. Certain Representations of Kajiwara--Watatani C*-correspondences
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $R$ be a rational function with $\deg\geq 2$ and $X$ be its Julia set, its Fatou set, or the whole of the Riemann sphere. Suppose that $X$ is not empty. Then, the following are equivalent. (1) $C_0(X)$ is a Cartan subalgebra of ${\mathcal{O}}_R(X)$. (2) $X$ has no branch point of $R$.

Theorems & Definitions (133)

  • Theorem 1.1: Theorem \ref{['4.14']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark
  • Proposition 2.4
  • proof
  • Proposition 2.6
  • proof
  • proof
  • ...and 123 more