Journal of Operator Theory

Volume 95, Issue 2, Spring 2026 pp. 339-369.

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

Authors: Kei Ito
Author institution: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Tokyo, 153-8914, Japan

Summary: Let $R$ be a rational function with degree $\geqslant 2$ 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~$\mathcal O_R(X)$.

DOI: http://dx.doi.org/10.7900/jot.2024apr09.2471
Keywords: Cartan subalgebra, $C^*$-algebra, complex dynamical system, branch\break point

Contents Full-Text PDF