Noncommutative Lp structure encodes exactly Jordan structure

We prove that for all 1⩽p⩽∞, p≠2, the Lp spaces associated to two von Neumann algebras M, N are isometrically isomorphic if and only if M and N are Jordan *-isomorphic. This follows from a noncommutative Lp Banach-Stone theorem: a specific decomposition for surjective isometries of noncommutative Lp spaces.