Canonical operator space structures on non-commutative Lp spaces

We analyze canonical operator space structures on the non-commutative L-P spaces L-n(p)(M; phi, omega) constructed by interpolation a la Stein-Weiss based on two normal semifinite faithful weights phi, omega on a W*-algebra M. We show that there is only one canonical (i.e. arising by interpolation) operator space structure on L-p(M) when M and p are kept fixed. Namely, for any n.s.f, weights phi, omega on M and eta epsilon [0, 1], the spaces L-eta(p)(M; phi, omega) are all completely isomorphic when they are canonically considered as operator spaces. Finally, we also describe the norms on all matrix spaces M-n(L-p(M)) which determine such a canonical quantized structure. (C) 1999 Academic Press.