The Space of Operator Valued Functions Seen as Hilbert H∗-Module

Abstract

Let M be a space of weakly ∗-measurable functions F: Ω → B(H) on measure space (Ω, Σ, µ), for which the function F∗F is Gel’fand integrable and Gel’fand integral R Ω F∗F dµ is a nuclear operator on Hilbert space H. We show that M is a Hilbert H∗-module, which contains an orthonormal basis.