  We prove that for all 1 \le p \le \infty, p not 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.
