Extendable endomorphisms on factors

Abstract

We begin this note with a von Neumann algebraic version of the elementary but extremely useful fact about being able to extend inner-product preserving maps from a total set of the domain Hilbert space to an isometry defined on the entire domain. This leads us to the notion of when `good' endomorphisms of a factorial probability space $(M,\phi)$ (which we call equi-modular) admit a natural extension to endomorphisms of $L^2(M,\phi)$. We exhibit examples of such extendable endomorphisms. We then pass to $E_0$-semigroups $\alpha = {\alpha_t: t \geq 0}$ of factors, and observe that extendability of this semigroup (i.e., extendability of each $\alpha_t$) is a cocycle-conjugacy invariant of the semigroup. We identify a necessary condition for extendability of such an $E_0$-semigroup, which we then use to show that the Clifford flow on the hyperfinite $II_1$ factor is not extendable.