Strong Operator Modules and the Haagerup Tensor Product

Abstract

Strong operator modules over a von Neumann algebra R are introduced and the so-called extended Haagerup tensor product over R of strong modules is studied. In the case R=C and the spaces are weak* closed this product agrees with the weak* Haagerup tensor product of Blecher and Smith. If C is the center of R, the extended Haagerup tensor product R is a Banach algebra containing the central tensor product R \haagc R of Chatterjee and Smith and has very nice properties concerning slice maps and the relative commutants of subalgebras. It is shown that R is a dual Banach space, and all weak* closed two-sided ideals of the algebra R are determined. 1991 Mathematics Subject Classification: 46L05