Real Interpolation between Row and Column Spaces
Volume 59 / 2011
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 of 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 norm.
