Extendable endomorphisms on factors

الأصل · EN

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,ϕ) (which we call equi-modular) admit a natural extension to endomorphisms of L²(M,ϕ). We exhibit examples of such extendable endomorphisms. We then pass to E₀-semigroups α= αₜ: t ≥ 0 of factors, and observe that extendability of this semigroup (i.e., extendability of each αₜ) is a cocycle-conjugacy invariant of the semigroup. We identify a necessary condition for extendability of such an E₀-semigroup, which we then use to show that the Clifford flow on the hyperfinite II₁ factor is not extendable.