On $C^{\ast }$-algebras associated to transfer operators for countable-to-one maps

Abstract

Our initial data is a transfer operator $L$ for a continuous, countable-to-one map $\phi :\Delta \to X$ defined on an open subset of a locally compact Hausdorff space $X$. Then $L$ may be identified with a `potential', i.e. a map $\varrho :\Delta \to X$ that need not be continuous unless $\phi$ is a local homeomorphism. We define the crossed product $C_0(X)⋊L$ as a universal $C^{\ast }$-algebra with explicit generators and relations, and give an explicit faithful representation of $C_0(X)⋊L$ under which it is generated by weighted composition operators. We explain its relationship with Exel-Royer's crossed products, quiver $C^{\ast }$-algebras of Muhly and Tomforde, $C^{\ast }$-algebras associated to complex or self-similar dynamics by Kajiwara and Watatani, and groupoid $C^{\ast }$-algebras associated to Deaconu-Renault groupoids. We describe spectra of core subalgebras of $C_0(X)⋊L$ and use it to characterise simplicity of $C_0(X)⋊L$ and prove the uniqueness theorem for $C_0(X)⋊L$. We give efficient criteria for $C_0(X)⋊L$ to be a Kirchberg algebra, and we discuss relationship between KMS states on the core subalgebra of $C_0(X)⋊L$ and conformal measures for $\phi$.

38 pages