Real Interpolation between Row and Column Spaces
Volume 59 / 2011
Abstract
We give an equivalent expression for the -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.