K-theory of noncommutative Bernoulli shifts

For a large class of C ∗ -algebras A , we calculate the K -theory of reduced crossed products A ⊗ G ⋊ 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.