Isometries on Noncommutative Symmetric Spaces

Abstract

Let $\mathcal{M}$ be an atomless semifinite von Neumann algebra equipped with a faithful normal semifinite trace τ (or else, an atomic von Neumann algebra with all atoms having the same trace) acting on a separable Hilbert space $\mathcal{H}$. Let $E(\mathcal{M},\tau )$ be a separable symmetric space of τ-measurable operators, whose norm is not proportional to the Hilbert norm ||⋅||₂ on ${{L}_{2}}(\mathcal{M},\tau )$. We provide a description of all bounded Hermitian operators on $E(\mathcal{M},\tau )$ and all surjective linear isometries of this space.