Canonical Operator Space Structures on Non-Commutative Lp Spaces

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