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

TL;DR

This paper introduces $C^*$-algebras linked to algebraic correspondences on the Riemann sphere, generalizing dynamical systems, and proves their simplicity and pure infiniteness under certain conditions.

Contribution

It defines $C^*$-algebras for algebraic correspondences on the Riemann sphere and establishes their properties when the correspondence is free and expansive.

Findings

The associated $C^*$-algebra ${ mf O}_p(J)$ is simple.

The algebra is purely infinite.

Results apply to free and expansive algebraic correspondences.

Abstract

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.