A Maurey type result for operator spaces

TL;DR

This paper establishes an operator space analogue of Maurey's theorem, showing that completely bounded maps from the compact operators to the operator Hilbert space are summing for any q>2 and admit a specific factorization.

Contribution

It introduces a Maurey-type result for operator spaces, demonstrating summing properties and factorizations for cb-maps into the operator Hilbert space.

Findings

Every cb-map from a0K to OH is (q,cb)-summing for q>2.

Such maps admit a factorization involving Schatten class operators.

The result extends classical Banach space theorems to the operator space setting.

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 $S_{2q}$.