Norms of skew-adjoint derivations with values in the predual of a semifinite von Neumann algebra

Abstract

A classical result due to Stampfli states that for any a∈B(H), the norm of the inner derivation δa(⋅)=[a,⋅] on B(H) is given by‖δa‖B(H)→B(H)=2inf⁡{‖a−c1‖∞:c∈C}. We establish a predual version of Stampfli's result: given a semifinite von Neumann algebra M equipped with a semifinite faithful normal trace τ, we show that for any self-adjoint a∈L1(M,τ),‖δa‖M→L1(M,τ)=2inf⁡{‖a−z‖1:z∈Z(LS(M))}, where Z(LS(M)) stands for the center of the space of all locally measurable operators affiliated with M, and ‖⋅‖1 stands for the norm of the noncommutative L1-space L1(M,τ) affiliated with M.