A Maurey type result for operator spaces

Abstract

The little Grothendieck theorem for Banach spaces says that every bounded linear operator between $C(K)$ and $\ell_2$ is 2-summing. However, it is shown in \cite{J05} that the operator space analogue fails. Not every cb-map $v : \K \to OH$ is completely 2-summing. In this paper, we show an operator space analogue of Maurey's theorem : Every cb-map $v : \K \to OH$ is $(q,cb)$-summing for any $q>2$ and hence admits a factorization $\|v(x)\| \leq c(q) \|v\|_{cb} \|axb\|_q$ with $a,b$ in the unit ball of the Schatten class $S_{2q}$.