A Maurey type result for operator spaces

The little Grothendieck theorem for Banach spaces says that every bounded linear operator between C(K) and ℓ2 is 2-summing. However, it is shown in [M. Junge, Embedding of the operator space OH and the logarithmic ‘little Grothendieck inequality’, Invent. Math. 161 (2) (2005) 225–286] that the operator space analogue fails. Not every cb-map is completely 2-summing. In this paper, we show an operator space analogue of Maurey's theorem: every cb-map is (q,cb)-summing for any q>2 and hence admits a factorization ‖v(x)‖⩽c(q)‖v‖cb‖axb‖q with a,b in the unit ball of the Schatten class S2q.