Non-Cocycle-Conjugate E${}_0$-Semigroups on Factors

Abstract

We investigate E${}_0$-semigroups on general factors, which are not necessarily of type I or II${}_1$. We show several families on the hyperfinite II${}_{\infty }$ factor, which arises as tensor products, consists of mutually non-cocycle-conjugate E${}_0$-semigroups. Using CCR representations associated with quasi-free states, we exhibit, for the first time, uncountable families consisting of mutually non-cocycle-conjugate E${}_0$-semigroups on all type III${}_{\lambda }$ factors, for $\lambda \in (0,1]$. They are not cocycle conjugate to the E${}_0$-semigroups constructed using CAR representations.