Remarks on the non-commutative Khintchine inequalities for $0<p<2$

Abstract: We show that the validity of the non-commutative Khintchine inequality for some $q$ with $1<q<2$ implies its validity (with another constant) for all $1\le p<q$. We prove this for the inequality involving the Rademacher functions, but also for more general "lacunary" sequences, or even non-commutative analogues of the Rademacher functions. For instance, we may apply it to the "Z(2)-sequences" previously considered by Harcharras. The result appears to be new in that case. It implies that the space $\ell^n_1$ contains (as an operator space) a large subspace uniformly isomorphic (as an operator space) to $R_k+C_k$ with $k\sim n^{\frac12}$. This naturally raises several interesting questions concerning the best possible such $k$. Unfortunately we cannot settle the validity of the non-commutative Khintchine inequality for $0<p<1$ but we can prove several would be corollaries. For instance, given an infinite scalar matrix $[x_{ij}]$, we give a necessary and sufficient condition for $[\pm x_{ij}]$ to be in the Schatten class $S_p$ for almost all (independent) choices of signs $\pm 1$. We also characterize the bounded Schur multipliers from $S_2$ to $S_p$. The latter two characterizations extend to $0<p<1$ results already known for $1\le p\le2$. In addition, we observe that the hypercontractive inequalities, proved by Carlen and Lieb for the Fermionic case, remain valid for operator space valued functions, and hence the Kahane inequalities are valid in this setting.