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 $R={\bf C}$ 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 \rcr\ 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 \rcr\ is a dual Banach space, and all weak* closed two-sided ideals of the algebra \rcr\ are determined.

1991 Mathematics Subject Classification: 46L05