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

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

2008-06-22

arxiv.org/abs/0806.3546v1

Mathematics

Operator Algebras

22 pages, LaTeX2e formated

Scientific paper

Let $p(z,w)$ be a polynomial in two variables. We call the solution of the algebraic equation $p(z,w) = 0$ the 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 $\hat{\mathbb C}$ give a generalization of dynamical systems of Klein 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 $\hat{\mathbb C}$, then the associated $C^*$-algebra ${\mathcal O}_p(J)$ is simple and purely infinite.

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