Cocycle conjugacy of quasifree endomorphisms semigroups on the CAR algebra

Abstract

W. Arveson has described a cocycle conjugacy class $U(\alpha)$ of $E_0$-semigroup $\alpha$ on B(H) which is a factor of type $\rm I$. Under some conditions on $\alpha$, there is a $E_0$-semigroup $\beta \in U(\alpha)$ being a flow of shifts in the sence of R.T. Powers. We study quasifree endomorphisms semigroups $\alpha$ on the hyperfinite factor $M=\pi (A(K))''$ generated by the representations $\pi$ of the algebra of canonical anticommutation relations A(K) over a separable Hilbert space K. The type of M can be $\rm I$, $\rm II$ or $\rm III$ depending on $\pi$. The cocycle conjugacy class $U(\alpha)$ is described in the terms of initial isometrical semigroups in K and an analogue of the Arveson result for the hyperfinite factors M of the type $\rm II_1$ and $\rm III_{\lambda}, 0<\lambda <1,$ is introduced.