A Computation with the Connes-Thom Isomorphism

Let A is an element of M-n(R) be an invertible matrix. Consider the semi-direct product R-n (sic) Z where the action of Z on R-n is induced by the left multiplication by A. Let (alpha, tau) be a strongly continuous action of R-n (sic) Z on a C*-algebra B where alpha is a strongly continuous action of R-n and tau is an automorphism. The map tau induces a map (tau) over tilde B (sic)(alpha) R-n. We show that, at the K-theory level, tau commutes with the Connes-Thom map if det(A) > 0 and anticommutes if det(A) < 0. As an application, we recompute the K-groups of the Cuntz-Li algebra associated with an integer dilation matrix.