On completely bounded bimodule maps over W$^*$-algebras
Volume 154 / 2003
Abstract
It is proved that for a von Neumann algebra $A\subseteq {\rm B}({\cal H})$ the subspace of normal maps is dense in the space of all completely bounded $A$-bimodule homomorphisms of ${\rm B}({\cal H})$ in the point norm topology if and only if the same holds for the corresponding unit balls, which is the case if and only if $A$ is atomic with no central summands of type $I_{\infty ,\infty }$. Then a duality result for normal operator modules is presented and applied to the following problem. Given an operator space $X$ and a von Neumann algebra $A$, is the map $q:A\mathbin {\mathrel {\mathop {\otimes }\limits ^{eh}}} X \mathbin {\mathrel {\mathop {\otimes }\limits ^{eh}}} A \to X\mathbin {\mathrel {\mathop {\otimes }\limits ^{np}}} A$, induced by $q(a\otimes x\otimes b)=x\otimes ab$, from the extended Haagerup tensor product to the normal version of the Pisier delta tensor product a quotient map? We give a reformulation of this problem in terms of normal extension of some completely bounded maps and answer it affirmatively in the case $A$ is of type I and $X$ belongs to a certain class which includes all finite-dimensional operator spaces.
