2003/12/11 by Javier Parcet
Mathematics · #math.OA #math.FA #msc:46L07 #msc:42C15
published as Proc. Edinburgh Math. Soc. 46 (2003), 649-668. · To appear in Proc. Edinburgh Math. Soc. 17 pages
arxiv created 2003/12/11 · arxiv updated 2009/12/01
The notion of B-convexity for operator spaces, which a priori depends on a set of parameters indexed by Σ, 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 BΣ-convex if and only if it has Σ-subtype. The class of uniformly non-L1(Σ) operator spaces, which is also the class of BΣ-convex operator spaces, is introduced. Moreover, an operator space having non-trivial Σ-type is BΣ-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 BΣ-convexity with respect to Σ is studied. This provides some interesting problems which will be posed.