Abstract
Using the Baum–Connes conjecture with coefficients, we develop a K-theory formula for reduced C*-algebras of strongly 0-E-unitary inverse semigroups, or equivalently, for a class of reduced partial crossed products. This generalizes and gives a new proof of previous K-theory results of Cuntz, Echterhoff and the author. Our K-theory formula applies to a rich class of C*-algebras which are generated by partial isometries. For instance, as new applications which could not be treated using previous results, we discuss semigroup C*-algebras of Artin monoids, Baumslag-Solitar monoids and one-relator monoids, as well as C*-algebras generated by right regular representations of semigroups of number-theoretic origin, and C*-algebras attached to tilings.
Highlights
Many prominent classes of C*-algebras, which played an important role in the development of the subject, are generated by partial isometries, for example AF algebras [5,20], Cuntz-Krieger algebras [16], graph algebras [47], tiling C*-algebras [26] or semigroup C*-algebras [15], to mention just a few
The notions of inverse semigroups and inverse semigroup C*-algebras [21,46] provide a natural and powerful framework to study these C*-algebras and their properties. Another very general and powerful concept is given by partial dynamical systems and the corresponding partial crossed product construction, which highlights the underlying dynamics at the heart of many C*-algebra constructions, including the ones mentioned above
Our K-theory formula applies to C*-algebras of tilings and their inverse semigroups, which are interesting from the point of view of dynamics as well as physics
Summary
Many prominent classes of C*-algebras, which played an important role in the development of the subject, are generated by partial isometries, for example AF algebras [5,20], Cuntz-Krieger algebras [16], graph algebras [47], tiling C*-algebras [26] or semigroup C*-algebras [15], to mention just a few. The main result of this paper can be reformulated in terms of partial dynamical systems In this context, it provides a K-theory formula for reduced crossed products of partial dynamical systems which admit an invariant regular basis of compact open sets, in the sense of Definition 2.12. Even in the setting of [13,14], i.e., for global dynamical systems admitting an invariant regular basis of compact open sets, the proof in the present paper differs from the one in [13,14] because the Morita enveloping action is always (except in trivial cases) defined on a noncommutative C*-algebra. Sehnem for pointing out that the classification result in Corollary 4.1 covers the examples in [25, Corollaries 5.4 and 5.5]
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