Duality and Normal Parts of Operator Modules

For an operator bimodule $X$ over von Neumann algebras $A\subseteq \text{bh}$ and $B\subseteq \text{bk}$, the space of all completely bounded $A,B$-bimodule maps from $X$ into $\text{bkh}$, 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 $\text{nor}X$ is associated so that completely bounded $A,B$-bimodule maps from $X$ into normal operator bimodules factorize uniquely through $\text{nor}X$. A construction of $\text{nor}X$ 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.