Regular normed bimodules

In this article, we will give a characterization of Banach bimodules over C∗-algebras of compact operators that arises from operator spaces as well as a characterization of (F)-Banach bundles amongst all (H)-Banach bundles over a hyper-Stonian space. These two characterizations are concerned with whether certain natural map from a Banach bimodule to its canonical bidual is isometric (we call such bimodule regular). 2000 Mathematics Subject Classification: 46H25, 46L07 Introduction The aim of this paper is to study duality theory for essential normed bimodules. Given a pre-C-algebra A and an essential normed A-bimodule X , we would like to have a canonical definition of the dual object X of X which satisfies the following properties: 1. X is also an essential normed A-bimodule (i.e. the dual object is in the same category); 2. X depends only on X and A; 3. when A = K(l) and X is defined by an operator space, X is defined by the corresponding dual operator space; 4. when A is commutative, X is the essential part of LA(X,A) (i.e. the duality agree with the usual one for commutative algebras). Let’s forget about the norm structure for the moment and consider a bimodule M over a unital algebra R. The natural “dual object” LR(M,R) fails to be a R-bimodule if R is not commutative. An easy way to rectify this situation is to “add another copy of R” and consider LR(M,R) where R is the algebraic tensor product R ⊙ R together with the R-bimodule structure: a · (b ⊗ c) · d = abd ⊗ c. Therefore, LR(M,R) becomes a R-bimodule (given by the bimodule structure on the second variable of R ⊙ R). However, when R is commutative, LR(M,R ⊙ R) 6= LR(M,R) unless R is the scalar field. A natural way to correct this is to replace R⊙R with R⊙Z R (where Z is the center of R). We employ this simple idea in Section 1 to define the “regular dual object”, X, of an essential normed A-bimodule X (for technical reason, we will assume that A has a contractive approximate identity and A = A). There is a canonical contraction κX : X → X ss (the dual of X). In general, κX is not an isometry and X is called regular if κX happens to be an isometry. It is easy to see that X is always regular and so, κX(X) is called the regularization of X . Regular bimodule are thought to be nice object because of the results in Sections 2 and 3. It is natural to ask whether one can give a This work is supported by the National Natural Science Foundation of China (10371058).