K-Theoretic Dynamics and Finiteness in C*-Crossed Products
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This work explores the interplay of C*-dynamics and K-theory. More precisely, we study the extent to which various forms of finite-dimensional approximation properties of a topological nature, witnessed in reduced C*-crossed products, are reflections of approximation conditions at the level of the dynamics. Such conditions admit purely algebraic K-theoretical interpretations that we describe and utilize to prove deep structural results. We introduce the notions of Matricial Field (MF) and Residually Finite Dimensional (RFD) actions of a discrete group Г on an arbitrary C*-algebra A. These actions have spatial interpretations in the case where the algebra A = C(X) is commutative; these are described. We show that a reduced crossed product Ax|ʎ Г is MF (RFD) if and only if the reduced group C*-algebra C*/ʎ(Г) is MF (RFD), and the action is MF (RFD). Examples include the limit periodic actions defined by Voiculescu and, in the classical case, the chain recurrent Z-systems of Pimsner. In the presence of sufficiently many projections MF and RFD actions can be expressed by elegant, simple, K-theoretic conditions. We then focus on 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. We also exhibit crossed product algebras for which the Ext semigroup is not a group; indeed any action of a free group on a UHF algebra gives rise to an MF crossed product whose Ext semigroup is not a group. Minimal C*-systems (A, Г) are described by certain filling conditions witnessed at the level of the induced actions of Г on K0(A) and on the Cuntz semigroup W(A). A notion of topological transitivity is defined for noncommutative systems again in terms of the induced action on K-theory. We prove that prime reduced crossed products come from topological transitive actions and, conversely, topologically transitive and properly outer systems yield prime reduced crossed products. In the presence of sufficiently many projections we associate to each noncommutative C*-system (A, Г, α) a type semigroup S(A, Г, 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 semi-group. Using ideas of paradoxical decompositions we obtain, for a certain class of simple crossed products, a dichotomy between the stably finite and purely infinite.
Rainone, Timothy (2015). K-Theoretic Dynamics and Finiteness in C*-Crossed Products. Doctoral dissertation, Texas A & M University. Available electronically from