The free shift $\alpha$ of the reduced free product $C^*$-algebra $A$ is studied from both the analytic and non-commutative ergodic theoretic viewpoints. For an automorphism $\beta$ of $B$, we show that the entropy of $\mathop{\rm Ad}\nolimits u(\alpha \otimes \beta)$ is equal to the entropy of $\mathop{\rm Ad}\nolimits u(\beta)$. We also show that if $B$ is unital, nuclear, and simple, and if the crossed product $B \rtimes_\beta {\Bbb Z}$ is simple and purely infinite, then $(O_\infty \otimes B)\rtimes_{\alpha \otimes \beta} {\Bbb Z}$ is isomorphic to $B \rtimes_\beta {\Bbb Z}$.

An action α of a discrete group Γ on the circle S 1 as orientation preserving C ∞-diffeomorphisms gives rise to a foliation on the homotopy quotient S 1Γ, and its Godbillon-Vey invariant is, by definition, a cohomology class of S 1Γ([1]). This cohomology class naturally defines an additive map from the geometric K-group K 0(S 1, Γ) into C, through the Chern character from K 0(S 1, Γ) to H *(S 1, Γ Q).

Using cyclic cohomology, Connes constructed in [2] an additive map, GV(α), which we shall call the Godbillon-Vey map, from the K 0-group of the reduced crossed product C*-algebra C(S 1) ⋊ αΓ into C. He showed that GV(α) agrees with the geometric Godbillon-Vey invariant through the index map μ from K 0(S 1, Γ) to K 0(C(S 1) ⋊ αΓ).