Journal of Operator Theory

Volume 65, Issue 2, Spring 2011 pp. 427-449.

$C^*$-algebras associated with algebraic correspondences on the Riemann sphere

Authors: Tsuyoshi Kajiwara (1) and Yasuo Watatani (2)
Author institution: (1) Department of Environmental and Mathematical Sciences, Okayama University, Tsushima, 700-8530, Japan
(2) Department of Mathematical Sciences, Kyushu University, Hakozaki, Fukuoka, 812-8581, Japan

Summary: Let $p(z,w)$ be a polynomial in two variables. We call the solution of the algebraic equation $p(z,w) = 0$ an algebraic correspondence. We regard it as the graph of the multivalued function $z \mapsto w$ defined implicitly by $p(z,w) = 0$. Algebraic correspondences on the Riemann sphere $\widehat{\mathbb C}$ generalize both Kleinian groups and rational functions. We introduce $C^*$-algebras associated with algebraic correspondences on the Riemann sphere. We show that if an algebraic correspondence is free and expansive on a closed $p$-invariant subset $J$ of $\widehat{\mathbb C}$, then the associated $C^*$-algebra ${\mathcal O}_p(J)$ is simple and purely infinite.

Keywords: algebraic correspondence, complex dynamical system, purely infinite $C^*$-algebra, Hilbert $C^*$-bimodule

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