$K$-theory of non-commutative Bernoulli Shifts

TL;DR

This paper computes the K-theory of reduced crossed products of Bernoulli shifts for a broad class of C*-algebras and groups, providing explicit formulas and applications to wreath product groups.

Contribution

It introduces a method to calculate K-theory for crossed products of Bernoulli shifts, extending previous techniques and applying to various algebra classes and wreath products.

Findings

Explicit K-theory formulas for finite-dimensional C*-algebras, UHF-algebras, and rotation algebras.

K-theory computation for reduced C*-algebras of wreath products.

Application of generalized techniques and trivialization theorems to broad classes of groups and algebras.

Abstract

For a large class of C*-algebras $A$, we calculate the $K$-theory of reduced crossed products $A^{\otimes G}\rtimes_rG$ 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 $H\wr G$ 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.