Martingale inequalities and operator space structures on $L_p$

Abstract

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

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.