This is joint work with Éric Ricard. We give a proof of the Khintchine inequalities in non- commutative $L_p$-spaces for all $0 < p < 1$. This case remained open since the first proof given by Francoise Lust-Piquard in 1986 for $1 < p < \infty$. These inequalities are valid for the Rademacher functions or Gaussian random variables, but also for more general sequences, e.g. for lacunary Fourier series or the analogues of Gaussian variables in free probability.

The Khintchine inequalities for non-commutative $L_p$-spaces play an important roˆle in the recent developments in non-commutative Functional Analysis, and in particular in Operator Space Theory. Just like their commutative counterpart for ordinary $L_p$-spaces, they are a crucial tool to understand the behavior of unconditionally convergent series of random variables, or random vectors, in non-commutative $L_p$. The commutative version for $p = 1$ is closely related to Grothendieck's Theorem. In the most classical setting, the non-commutative Khintchine inequalities deal with Rademacher series of the form

$S=\sum_kr_k(t)x_k$

where $(r_k)$ are the Rademacher functions on the Lebesgue interval where the coefficients $x_k$ are in the Schatten $q$-class or in a non-commutative $L_q$-space associated to a semifinite trace $\tau$. Let us denote simply by $||.||_q$ the norm (or quasi-norm) in the latter Banach (or quasi-Banach) space, that we will denote by $L_q(\tau)$. When $\tau$ is the usual trace on $B(\ell_2)$, we recover the Schatten $q$-class. By Kahane's well known results, $S$ converges almost surely in norm if it converges in $L_q(dt;L_q(\tau))$. Thus to characterize the almost sure norm-convergence for series such as $S$, it suffices to produce a two sided equivalent of $||S||_{L_q(dt;L_q(\tau))}$ when $S$ is a finite sum, and this is precisely what the non-commutative Khintchine inequalities provide :
For any $0 < q < \infty$ there are positive constants $\alpha_q,\beta_q$ such that for any finite set $(x_1, . . . , x_n)$ in $L_q(\tau)$ we have

$(\beta_q)^{-1}|||(x_k)|||_q\leq\left(\int||S(t)||^q_qdt\right)^{1/q}\leq\alpha_q|||(x_k)|||_q$

where $|||(x_k)|||_q$ is defined as follows :
If $2\le q<\infty$

$|||x_k|||_q \overset{def}{=} \max\lbrace ||(\sum x^*_k x_k)^{1/2} ||_q, ||(\sum x_kx^*_k)^{1/2}||_q\rbrace$ (1)

and if $0\le q<2$:

$|||x|||_q \overset{def}{=} \underset{x_k=a_k+b_k}{inf} \lbrace ||(\sum a^*_ka_k)^{1/2} ||_q + ||(\sum b_kb^*_k)^{1/2}||_q\rbrace$. (2)

Note that $\beta=1$ if $q\ge2$, while $\alpha_q=1$ if $q\le2$ and the corresponding one sided bounds are easy. The difficulty is to verify the other side.[-]