Notes on real interpolation of operator $L_p$-spaces

TL;DR

This paper investigates the structure of noncommutative Lp-spaces associated with semifinite von Neumann algebras, revealing conditions under which certain interpolation and isomorphism properties hold, and establishing inequalities related to these spaces.

Contribution

It proves that the real interpolation space L_{p,p}(rac{1}{p},p) coincides with L_p(rac{1}{p}) only for finite-dimensional algebras, and characterizes when certain vector-valued spaces are isomorphic to classical Lp-spaces.

Findings

L_{p,p}(rac{1}{p},p) = L_p(rac{1}{p}) iff rac{1}{p} is finite-dimensional.

Vector-valued interpolation spaces are isomorphic to classical Lp-spaces iff the algebra is commutative.

Established inequalities for sums of positive elements in noncommutative Lp-spaces depending on algebra structure.

Abstract

Let $\mathcal{M}$ be a semifinite von Neumann algebra. We equip the associated noncommutative $L_p$-spaces with their natural operator space structure introduced by Pisier via complex interpolation. On the other hand, for $1<p<\infty$ let $$L_{p,p}(\mathcal{M})=\big(L_{\infty}(\mathcal{M}),\,L_{1}(\mathcal{M})\big)_{\frac1p,\,p}$$ be equipped with the operator space structure via real interpolation as defined by the second named author ({\em J. Funct. Anal}. 139 (1996), 500--539). We show that $L_{p,p}(\mathcal{M})=L_{p}(\mathcal{M})$ completely isomorphically if and only if $\mathcal{M}$ 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 $1<p<\infty$ and $1\le q\le\infty$ with $p\neq q$ $$\big(L_{\infty}(\mathcal{M};\ell_q),\,L_{1}(\mathcal{M};\ell_q)\big)_{\frac1p,\,p}=L_p(\mathcal{M}; \ell_q)$$ with equivalent norms, i.e., at the Banach space level if and only if $\mathcal{M}$ is isomorphic, as a Banach space, to a commutative von Neumann algebra. Our third result concerns the following inequality: $$ \big\|\big(\sum_ix_i^q\big)^{\frac1q}\big\|_{L_p(\mathcal{M})}\le\big\|\big(\sum_ix_i^r\big)^{\frac1r}\big\|_{L_p(\mathcal{M})} $$ for any finite sequence $(x_i)\subset L_p^+(\mathcal{M})$, where $0<r<q<\infty$ and $0<p\le\infty$. If $\mathcal{M}$ is not isomorphic, as a Banach space, to a commutative von Meumann algebra, then this inequality holds if and only if $p\ge r$.