It is proved that for a von Neumann algebra A ⊆ B(ℋ ) the subspace of normal maps is dense in the space of all completely bounded A-bimodule homomorphisms of B(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 . 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 , induced by q(a ⊗ x ⊗ b) = x ⊗ 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm154-2-3,
author = {Bojan Magajna},
title = {On completely bounded bimodule maps over W*-algebras},
journal = {Studia Mathematica},
volume = {157},
year = {2003},
pages = {137-164},
zbl = {1023.46065},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm154-2-3}
}
Bojan Magajna. On completely bounded bimodule maps over W*-algebras. Studia Mathematica, Tome 157 (2003) pp. 137-164. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm154-2-3/