COMPLETE ISOMETRIES BETWEEN SUBSPACES OF NONCOMMUTATIVE Lp-SPACES

We prove some noncommutative analogues of a theorem proved by Plotkin and Rudin about isometries between subspaces of L-p-spaces. Let 0 < p < infinity, p not an even integer. The main result of this paper states that in the category of unital subspaces of noncommutative probability L-p-spaces, under some boundedness condition, the unital completely isometric maps come from *-isomorphisms of the underlying von Neumann algebras. Some applications are given, including to noncommutative H-p-spaces.