We analyze canonical operator space structures on the non-commutative L-P s
paces L-n(p)(M; phi, omega) constructed by interpolation a la Stein-Weiss b
ased 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-et
a(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 struct
ure. (C) 1999 Academic Press.