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: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.