  In this article we show that positive surjective isometries between symmetric
spaces associated with semi-finite von Neumann algebras are projection
disjointness preserving if they are finiteness preserving. This is subsequently
used to obtain a structural description of such isometries. Furthermore, it is
shown that if the initial symmetric space is a strongly symmetric space with
absolutely continuous norm, then a similar structural description can be
obtained without requiring positivity of the isometry.
