Notes on Real Interpolation of Operator Lp-Spaces

Abstract

Let $${\cal M}$$ 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 $${L_{p,p}}({\cal M}) = {({L_\infty}({\cal M}),\,{L_1}({\cal M}))_{{1 \over p},p}}$$ 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 $${L_{p,p}}({\cal M}) = {L_p}({\cal M})$$ completely isomorphically if and only if $${\cal M}$$ 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 $${({L_\infty}({\cal M};{\ell _q}),\,\,{L_1}({\cal M};{\ell _q}))_{{1 \over p},p}} = {L_p}({\cal M};{\ell _q})$$ with equivalent norms, i.e., at the Banach space level if and only if $${\cal M}$$ is isomorphic, as a Banach space, to a commutative von Neumann algebra. Our third result concerns the following inequality: $${\left\| {{{(\sum\limits_i {x_i^q})}^{{1 \over q}}}} \right\|_{{L_p}({\cal M})}} \le {\left\| {{{(\sum\limits_i {x_i^r})}^{{1 \over r}}}} \right\|_{{L_p}({\cal M})}}$$ for any finite sequence $$({x_i}) \subset L_p^ + ({\cal M})$$ , where 0 < r < q < ∞ and 0 < p ≤ ∞. If $${\cal M}$$ is not isomorphic, as a Banach space, to a commutative von Meumann algebra, then this inequality holds if and only if p ≥ r.