We study the finite versus infinite nature of C * -algebras arising frométale groupoids. For an ample groupoid G, we relate infiniteness of the reduced C * -algebra of G to notions of paradoxicality of a K-theoretic flavor. We construct a pre-ordered abelian monoid S(G) which generalizes the type semigroup introduced by Rørdam and Sierakowski for totally disconnected discrete transformation groups. This monoid reflects the finite/infinite nature of the reduced groupoid C * -algebra of G. If G is ample, minimal, and topologically principal, and S(G) is almost unperforated we obtain a dichotomy between stable finiteness and pure infiniteness for the reduced C * -algebra of G.
We discuss the interplay between K-theoretical dynamics and the structure theory for certain C * -algebras arising from crossed products. For noncommutative C *systems we present notions of minimality and topological transitivity in the K-theoretic framework which are used to prove structural results for reduced crossed products. In the presence of sufficiently many projections we associate to each noncommutative C * -system (A, G, α) a type semigroup S(A, G, α) which reflects much of the spirit of the underlying action. We characterize purely infinite as well as stably finite crossed products by means of the infinite or rather finite nature of this semigroup. We explore the dichotomy between stable finiteness and pure infiniteness in certain classes of reduced crossed products by means of paradoxical decompositions. TIMOTHY RAINONEfinite projection [31]. The conjecture for such a dichotomy remains open for those algebras whose projections are total. Theorem 1.3 below is a result in this direction.Despite the failure of the above dichotomy, the classification program of Elliott in its original K-theoretic formulation has witnessed much success for stably finite algebras [32], [13], as well as in the purely infinite case with the spectacular complete classification results of Kirchberg and Phillips [27], [20] modulo the UCT. One motivation for studying purely infinite algebras stems from the fact that Kirchberg algebras (unital, simple, separable, nuclear, and purely infinite) are classified by their K-or KK-theory. The C * -literature has produced examples of purely infinite C * -algebras arising from dynamical systems [4], [22], [23], [33]. In many cases the underlying algebra is abelian with spectrum the Cantor set. For example, Archbold, Spielberg, and Kumjian (independently) proved that there is an action of Z 2 * Z 3 on the Cantor set so that the corresponding crossed product C * -algebra is isomorphic to O 2 [34]. Laca and Spielberg [23] construct purely infinite and simple crossed products that emerge from strong boundary actions. Jolissaint and Robertson [17] generalized the idea of strong boundary action to noncommutative systems with the concept of an n-filling action. They showed that A ⋊ λ Γ is simple and purely infinite provided that the action is properly outer and n-filling and every corner pAp of A is infinite dimensional. When the algebra A has a well behaved K 0 (A) group we will in fact give a K-theoretic proof of their result (see Proposition 4.20).The transition from classical topological dynamics to noncommutative C * -dynamics presents several challenges and subtleties. One way to approach these issues is to interpret dynamical conditions K-theoretically via the induced actions on K 0 (A) and on the Cuntz semigroup W (A) and use tools from the classification literature as well as developed techniques of Cuntz comparison to uncover pertinent algebraic information. Such an approach is seen in Brown's work [9] as well as that of the author in [29]. We continue this philosophy here. For instanc...
We study the interplay of C * -dynamics and K-theory. Notions of chain recurrence for transformations groups (X, Γ) and MF actions for non-commutative C * -dynamical systems (A, Γ, α) are translated into K-theoretical language, where purely algebraic conditions are shown to be necessary and sufficient for a reduced crossed product to admit norm microstates. We are particularly interested in actions of free groups on AF algebras, in which case we prove that a K-theoretic coboundary condition determines whether or not the reduced crossed product is a Matricial Field (MF) algebra. One upshot is the equivalence of stable finiteness and being MF for these reduced crossed product algebras.
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