  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 with
analogous results.
