Isometries on algebras of locally measurable operators

Abstract

Let $LS(\mathcal{M})$ be the algebra of locally measurable operators affiliated with a von Neumann algebra $\mathcal{M}$, equipped with an $F$-norm defined via a dimension function and a probability measure. We prove that every bijective linear isometry between $LS(\mathcal{M})$ admits a canonical representation of the form $\Phi (x)=wJ(x)$, where $w$ is a unitary element and $J$ is a Jordan ${}^{\ast }$-isomorphism, which extends classical results such as the Banach--Stone theorem and Kadison's theorem. Under several structural assumptions on the underlying von Neumann algebras (including all type ${\mathrm{II}}_{\infty }$ and type $\mathrm{III}$ algebras, and all factors, and algebras with atomless centers), we prove the one-to-one correspondence between the $F$-norm and the pair $(\mu ,D)$ of a probability measure and a dimension function, which fails for algebras with atomic centers.