Extendable endomorphisms on factors

TL;DR

This paper explores conditions under which certain endomorphisms and semigroups on factors can be extended to larger spaces, providing examples and invariants relevant to operator algebra theory.

Contribution

It introduces the concept of extendability for endomorphisms and $E_0$-semigroups on factors, and demonstrates its invariance and limitations through examples.

Findings

Extendability is a cocycle-conjugacy invariant.

Clifford flow on the hyperfinite II_1 factor is not extendable.

Provides a necessary condition for extendability.

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.