Strong Operator Modules and the Haagerup Tensor Product

Strong operator modules over a von Neumann algebra R are introducedand the so-called extended Haagerup tensor product over R ofstrong modules is studied. In the case R=C and the spaces areweak* closed this product agrees with the weak* Haagerup tensorproduct of Blecher and Smith. If C is the center of R, the extendedHaagerup tensor product R is a Banach algebra containing thecentral tensor product of Chatterjee and Smith and has very nice properties concerningslice maps and the relative commutants of subalgebras. It isshown that R is a dual Banach space, and all weak* closed two-sidedideals of the algebra R are determined. 1991 Mathematics SubjectClassification: 46L05