C∗-Algebras Associated to Transfer Operators for Countable-to-One Maps.

Autor: Bardadyn, Krzysztof, Kwaśniewski, Bartosz K., Lebedev, Andrei V.
Zdroj: Integral Equations & Operator Theory; Sep2024, Vol. 96 Issue 3, p1-45, 45p
Abstrakt: Our initial data is a transfer operator L for a continuous, countable-to-one map φ : Δ → 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 ϱ : Δ → X that need not be continuous unless φ is a local homeomorphism. We define the crossed product C 0 (X) ⋊ L as a universal C ∗ -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 ∗ -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 C 0 (X) ⋊ L , prove uniqueness theorems for C 0 (X) ⋊ L and characterize simplicity of C 0 (X) ⋊ L . We give efficient criteria for C 0 (X) ⋊ L to be purely infinite simple and in particular a Kirchberg algebra. [ABSTRACT FROM AUTHOR]
Databáze: Complementary Index
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