Real interpolation between row and column spaces

TL;DR

This paper characterizes the real interpolation spaces between operator row and column spaces, extends results to non-commutative Lorentz spaces, and proves a non-commutative Khintchine inequality valid for a broad range of parameters.

Contribution

It provides an equivalent expression for the K-functional between row and column operator spaces and extends non-commutative Khintchine inequalities to Lorentz spaces for all p,q in [1,∞).

Findings

Equivalent description of real interpolation spaces for operator spaces.

Extension of non-commutative Khintchine inequalities to Lorentz spaces.

Validation of a simultaneous decomposition property for operator norms.

Abstract

We give an equivalent expression for the $K$-functional associated to the pair of operator spaces $(R,C)$ formed by the rows and columns respectively. This yields a description of the real interpolation spaces for the pair $(M_n(R), M_n(C))$ (uniformly over $n$). More generally, the same result is valid when $M_n$ (or $B(\ell_2)$) is replaced by any semi-finite von Neumann algebra. We prove a version of the non-commutative Khintchine inequalities (originally due to Lust--Piquard) that is valid for the Lorentz spaces $L_{p,q}(\tau)$ associated to a non-commutative measure $\tau$, simultaneously for the whole range $1\le p,q< \infty$, regardless whether $p<2 $ or $p>2$. Actually, the main novelty is the case $p=2,q\not=2$. We also prove a certain simultaneous decomposition property for the operator norm and the Hilbert-Schmidt one.