2023/03/15 by Duquet, Charles, Merdy, Christian Le · 1 citation
#46E40 #46L06 #47A20 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2303.08436
Let M be a von Neumann algebra equipped with a normal semi-finite faithful trace (nsf trace in short) and let T\colon M→ M be a contraction. We say that T is absolutely dilatable if there exist another von Neumann algebra M' equipped with a nsf trace, a w^*-continuous trace preserving unital *-homomorphim J\colon M→ M' and a trace preserving *-automomorphim U\colon M'→ M' such that Tk=E Uk J for all integer k≥ 0, where E\colon M'→ M is the conditional expectation associated with J. Given a σ-finite measure space (Ω,μ), we characterize bounded Schur multipliers ϕ∈ L^∞(Ω2) such that the Schur multiplication operator Tϕ\colon B(L2(Ω))→ B(L2(Ω)) is absolutely dilatable. In the separable case, they are characterized by the existence of a von Neumann algebra N with a separable predual, equipped with a normalized normal faithful trace τN, and of a w^*-continuous essentially bounded function d\colonΩ→ N such that ϕ(s,t)=τN(d(s)^*d(t)) for almost every (s,t)∈Ω2.