K-theory of noncommutative Bernoulli shifts

Abstract

AbstractFor a large class of $$C^*$$ C ∗ -algebras A, we calculate the K-theory of reduced crossed products $$A^{\otimes G}\rtimes _rG$$ A ⊗ G ⋊ r G of Bernoulli shifts by groups satisfying the Baum–Connes conjecture. In particular, we give explicit formulas for finite-dimensional $$C^*$$ C ∗ -algebras, UHF-algebras, rotation algebras, and several other examples. As an application, we obtain a formula for the K-theory of reduced $$C^*$$ C ∗ -algebras of wreath products $$H\wr G$$ 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.