B-convex operator spaces

Details B-convex operator spaces B-convex operator spaces

2003-12-11

arxiv.org/abs/math/0312246v1

Proc. Edinburgh Math. Soc. 46 (2003), 649-668.

Mathematics

Operator Algebras

To appear in Proc. Edinburgh Math. Soc. 17 pages

Scientific paper

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.

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