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# Wydawnictwa / Czasopisma IMPAN / Bulletin Polish Acad. Sci. Math. / Wszystkie zeszyty

## Real Interpolation between Row and Column Spaces

### Tom 59 / 2011

Bulletin Polish Acad. Sci. Math. 59 (2011), 237-259 MSC: 46L07, 47L25, 46B70. DOI: 10.4064/ba59-3-6

#### Streszczenie

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.

#### Autorzy

• Gilles PisierMathematics Department
Texas A&M University
College Station, TX 77843, U.S.A.
and
Université Paris VI
Institut Mathématique de Jussieu
Analyse Fonctionnelle, Case 186
75252 Paris Cedex 05, France
e-mail

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