Real interpolation between row and column spaces

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 $p=2,q\ne 2$. We also prove a certain simultaneous decomposition property for the operator norm and the Hilbert-Schmidt one.