Let M be a semifinite von Neumann algebra.We equip the associated non-commutative Lp-spaces with their natural operator space structure introduced by Pisier via complex interpolation.On the other hand,for 1 ＜ p ＜ ∞ let Lp,p(M) =(L∞(M),L1 (M)) 1/qp,p be equipped with the operator space structure via real interpolation as defined by the second named author (J.Funct.Anal.139 (1996),500-539).We show that Lp,p(M) =Lp(M)completely isomorphically if and only if M is finite dimensional.This solves in the negative the three problems left open in the quoted work of the second author.We also show that for 1＜p＜∞and 1≤q≤∞wit hp≠q(L∞(M;(l)q),L1(M;(l)q)) 1 =Lp(M;(l)q)with equivalent norms,i.e.,at the Banach space level if and only if M is isomorphic,as a Banach space,to a commutative von Neumann algebra.Our third result concerns the following inequality:‖(Σixqi)1/q‖Lp(M) ≤‖(Σixqi)1/r‖ Lp(M)for any finite sequence (xi) C L+p(M),where 0 ＜ r ＜ q ＜ ∞ and 0 ＜ p ≤ ∞.If M is not isomorphic,as a Banach space,to a commutative von Meumann algebra,then this inequality holds if and only if p ≥ r.