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Real Interpolation between Row and Column Spaces

Pisier G.

Bulletin of the Polish Academy of Sciences. Mathematics

2011

Vol. 59, no 3

237--259

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 (Mn(R),Mn(C)) (uniformly over n). More generally, the same result is valid when Mn (or B(ℓ2)) 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 Lp,q(τ) associated to a non-commutative measure τ, simultaneously for the whole range 1≤p,q<∞, regardless of whether p<2 or p>2. Actually, the main novelty is the case p=2, q/=2. We also prove a certain simultaneous decomposition property for the operator norm and the Hilbert–Schmidt norm.