Notes on Real Interpolation of Operator Lp-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 L p , p ( ℳ ) = ( L ∞ ( 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 , 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 ; 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: ‖ ( ∑ i x i q ) 1 q ‖ L p ( ℳ ) ≤ ‖ ( ∑ i x i r ) 1 r ‖ L p ( ℳ ) for any finite sequence ( x i ) ⊂ 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 .