Notes on Real Interpolation of Operator L_p-Spaces

Let ℳ be a semifinite von Neumann algebra. We equip the associated non-commutative L_p-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))1p,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 L_p,p(ℳ) = L_p(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 ≤ ∞ with p ≠ q(L∞(ℳ;ℓq),L1(ℳ;ℓq))1p,p=Lp(ℳ;ℓq) 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: ‖(∑ixiq)1q‖Lp(ℳ)≤‖(∑ixir)1r‖Lp(ℳ) for any finite sequence (xi)⊂Lp+(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.