K-theory of non-commutative Bernoulli Shifts

Sayan Chakraborty, Siegfried Echterhoff, Julian Kranz, Shintaro Nishikawa

Abstract: For a large class of C*-algebras A, we calculate the K-theory of reduced crossed products $A^{\otimes G}\rtimes_r G$ of Bernoulli shifts by groups satisfying the Baum--Connes conjecture. In particular, we give explicit formulas for finite-dimensional C*-algebras, UHF-algebras, rotation algebras, and several other examples. As an application, we obtain a formula for the K-theory of reduced C*-algebras of wreath products H≀G for large classes of groups H and G. Our methods use a generalization of techniques developed by the second named author together with Joachim Cuntz and Xin Li, and a trivialization theorem for finite group actions on UHF algebras developed in a companion paper by the third and fourth named authors.