$C^{\ast }$-algebras associated with algebraic correspondences on the Riemann sphere

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 $C^{\ast }$-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 $C^{\ast }$-algebra ${\mathcal{O}}_p(J)$ is simple and purely infinite.

22 pages, LaTeX2e formated

References in corpus (3)

• Hilbert spaces built on a similarity and on dynamical renormalization
• C^*-algebras associated with self-similar sets
• C*-algebras associated with branched coverings