Trace preserving isomorphism on von Neumann algebras

Question

Is the condition on an isomorphism between von neumann algebras which says that the trace is conserved the same thing as the notion of spatial isomorphism?

Answer 1

Depends on how the question is interpreted. Any isomorphism between two tracial von Neumann algebras that preserves the trace extends to a unitary between the corresponding $L^2$ spaces. Conjugation by this unitary then gives a spatial isomorphism between these two tracial von Neumann algebras - if they are already in their standard forms to begin with (i.e., they are represented on their $L^2$ spaces via left regular representations). After all, spatial isomorphisms only make sense if you specify representations of the algebras involved. If they are not in their standard forms there's no reason why your isomorphism should be spatial.