\(K\)-theory of certain purely infinite crossed products (Q298135)

Let \(G\) be the free group on \(n\) generators, \(\beta G\) its Stone-Čech compactification, and let \(A=C(\beta G\setminus G)\) be the corresponding corona algebra. It is shown that there exist two \(G\)-invariant sub-\(C^*\)-algebras \(A_1,A_2\subset A\) such that \(A_1\rtimes_r G\) and \(A_2\rtimes_r G\) are Kirchberg algebras in the UCT class with different \(K\)-theory groups. It is also shown that \(G\) admits a free, amenable, minimal action on the Cantor set \(X\) such that \(C(X)\rtimes_r G\) is a Kirchberg algebra in the UCT class and \(K_0(C(X)\rtimes_r G)=0\).