2012/11/12 by Panchugopal Bikram, Masaki Izumi, Bikram, Panchugopal +5
Mathematics · #46L55 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.1211.2576
openalex publication_date 2012/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.