On $C^*$-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 ϕ : ∆ → 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 ̺ : ∆ → X that need not be continuous unless ϕ is a local homeomorphism. We define the crossed product C0(X) ⋊ L as a universal C * -algebra with explicit generators and relations, and give an explicit faithful representation of C0(X) ⋊ L under which it is generated by weighted composition operators. We explain its relationship with Exel-Royer's crossed products, quiver C * -algebras of Muhly and Tomforde, C * -algebras associated to complex or self-similar dynamics by Kajiwara and Watatani, and groupoid C * -algebras associated to Deaconu-Renault groupoids.We describe spectra of core subalgebras of C0(X) ⋊ L and use it to characterise simplicity of C0(X) ⋊ L and prove the uniqueness theorem for C0(X) ⋊ L. We give efficient criteria for C0(X) ⋊ L to be a Kirchberg algebra, and we discuss relationship between KMS states on the core subalgebra of C0(X) ⋊ L and conformal measures for ϕ.

Spectrum of weighted isometries: C*-algebras, transfer operators and topological pressure

“…For p = 2 the above definition agrees with the one in [11], see [13], [4] and Remark 6 below. For p = 2, as we explain in Section 2, covariant representations are given by multiplication operators and weighted composition operators.…”