B-convex operator spaces

Abstract

The notion of B-convexity for operator spaces, which a priori depends on a set of parameters indexed by $\Sigma$, is defined. Some of the classical characterizations of this geometric notion for Banach spaces are studied in this new context. For instance, an operator space is $B_{\Sigma}$-convex if and only if it has $\Sigma$-subtype. The class of uniformly non-$L^1(\Sigma)$ operator spaces, which is also the class of $B_{\Sigma}$-convex operator spaces, is introduced. Moreover, an operator space having non-trivial $\Sigma$-type is $B_{\Sigma}$-convex. However, the converse is false. The row and column operator spaces are nice counterexamples of this fact, since both are Hilbertian. In particular, this result shows that a version of the Maurey-Pisier theorem does not hold in our context. Some other examples of Hilbertian operator spaces will be treated. In the last part of this paper, the independence of $B_{\Sigma}$-convexity with respect to $\Sigma$ is studied. This provides some interesting problems which will be posed.