Duality and Normal Parts of Operator Modules

Details Duality and Normal Parts of Operator Modules Duality and Normal Parts of Operator Modules

2003-07-07

arxiv.org/abs/math/0307079v2

J.F.A. 219/2, pp. 306--339, February 2005

Mathematics

Operator Algebras

The first version of the paper has been split into two parts, corrected and a few results added. This is the first part

Scientific paper

For an operator bimodule $X$ over von Neumann algebras $A\subseteq\bh$ and $B\subseteq\bk$, the space of all completely bounded $A,B$-bimodule maps from $X$ into $\bkh$, is the bimodule dual of $X$. Basic duality theory is developed with a particular attention to the Haagerup tensor product over von Neumann algebras. To $X$ a normal operator bimodule $\nor{X}$ is associated so that completely bounded $A,B$-bimodule maps from $X$ into normal operator bimodules factorize uniquely through $\nor{X}$. A construction of $\nor{X}$ in terms of biduals of $X$, $A$ and $B$ is presented. Various operator bimodule structures are considered on a Banach bimodule admitting a normal such structure.

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