Duality and normal parts of operator modules

For an operator bimodule X over von Neumann algebras A⊆B(H) and B⊆B(K), the space of all completely bounded A,B-bimodule maps from X into B(K,H), 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 Xn is associated so that completely bounded A,B-bimodule maps from X into normal operator bimodules factorize uniquely through Xn. A construction of Xn 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.