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 < infinity let L-p,L-p(M) = L-infinity(M), L-1(M)(1/p, 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,L-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 < infinity and 1 <= q <= infinity with p not equal q (L-infinity(M; l(q)), L-1(M; l(q))(1/p, p) = L-p(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: parallel to(Sigma(i)x(i)(q))(1/q)parallel to(Lp (M)) <=parallel to(Sigma(i)x(i)(r))(1/r)parallel to(Lp (M)) for any finite sequence (x(i)) subset of L-p(+) (M), where 0 < r < q < infinity and 0 < p <= infinity. 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.\textbackslash