This is an announcement for the paper "Real interpolation between row and column spaces" by Gilles Pisier.

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.

Archive classification: math.OA

Mathematics Subject Classification: 47B10

Submitted from: pisier@math.jussieu.fr

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1109.1860

or

http://arXiv.org/abs/1109.1860