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

arXiv2303.14860MaRDI QIDQ6509347

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Abstract: Let $R$ be a rational function with degree $geq2$ 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 $mathcal{O}_R(X)$.

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