On completely bounded bimodule maps over W$^*$-algebras

Bojan Magajna¹

¹ Department of Mathematics University of Ljubljana Jadranska 19 Ljubljana 1000, Slovenia

Studia Mathematica, Tome 154 (2003) no. 2, pp. 137-164

Résumé

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.

Détail

DOI : 10.4064/sm154-2-3

Keywords: proved von neumann algebra subseteq cal subspace normal maps dense space completely bounded a bimodule homomorphisms cal point norm topology only holds corresponding unit balls which only atomic central summands type infty infty duality result normal operator modules presented applied following problem given operator space von neumann algebra map mathbin mathrel mathop otimes limits mathbin mathrel mathop otimes limits mathbin mathrel mathop otimes limits induced otimes otimes otimes extended haagerup tensor product normal version pisier delta tensor product quotient map reformulation problem terms normal extension completely bounded maps answer affirmatively type belongs certain class which includes finite dimensional operator spaces