Hermitian operators and isometries on symmetric operator spaces

Let M be an atomless semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbert space H equipped with a semifinite faithful normal trace . Let E.M; / be a symmetric operator space affiliated with M, whose norm is order continuous and is not proportional to the Hilbertian norm k.k₂ on L₂.M; /. We obtain a general description of all bounded hermitian operators on E.M; /. This is the first time that the description of hermitian operators on a symmetric operator space (even for a noncommutative L_p-space) is obtained in the setting of general (non-hyperfinite) von Neumann algebras. As an application, we resolve a long-standing open problem concerning the description of isometries raised in the 1980s, which generalizes and unifies numerous earlier results.