Geodesic
Martingale inequalities and operator space structures on $L_p$
Documenta mathematica, Tome 19 (2014), pp. 1367-1442.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We describe a new operator space structure on $L_p$ when $p$ is an even integer and compare it with the one introduced in our previous work using complex interpolation. For the new structure, the Khintchine inequalities and Burkholder's martingale inequalities have a very natural form:the span of the Rademacher functions is completely isomorphic to the operator Hilbert space $OH$, and the square function of a martingale difference sequence $d_n$ is $\Sigma d_n\otimes \bar d_n$. Various inequalities from harmonic analysis are also considered in the same operator valued framework. Moreover, the new operator space structure also makes sense for non-commutative $L_p$-spaces associated to a trace with analogous results. When $p\to \infty$ and the trace is normalized, this gives us a tool to study the correspondence $E\mapsto \underline{E}$ defined as follows: if $E\subset B(H)$ is a completely isometric emdedding then $ \underline{E}$ is defined so that $ \underline{E}\subset CB(OH)$ is also one.
Classification : 47L07, 46L53, 46B28, 60G48, 47L25 
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     author = {Pisier, Gilles},
     title = {Martingale inequalities and operator space structures on $L_p$},
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     volume = {19},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2014__19__a0/}
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Pisier, Gilles. Martingale inequalities and operator space structures on $L_p$. Documenta mathematica, Tome 19 (2014), pp. 1367-1442. http://geodesic.mathdoc.fr/item/DOCMA_2014__19__a0/