Cocycle conjugacy of quasifree endomorphisms semigroups on the CAR algebra

Abstract

W. Arveson has described a cocycle conjugacy class $U(α)$ of $E_0$-semigroup $α$ on B(H) which is a factor of type $\rm I$. Under some conditions on $α$, there is a $E_0$-semigroup $β\in U(α)$ being a flow of shifts in the sence of R.T. Powers. We study quasifree endomorphisms semigroups $α$ on the hyperfinite factor $M=π(A(K))&#39;&#39;$ generated by the representations $π$ 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 $π$. The cocycle conjugacy class $U(α)$ 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 $\rm III_λ, 0<λ<1,$ is introduced.