On completely bounded bimodule maps over W*-algebras

Bojan Magajna. "On completely bounded bimodule maps over W*-algebras." Studia Mathematica 154.2 (2003): 137-164. <http://eudml.org/doc/284437>.

@article{BojanMagajna2003,
abstract = {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 $I_\{∞,∞\}$. 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 $q:A \{⊗\limits ^\{eh\}\} X ⊗\{\limits ^\{eh\}\} A → X \{⊗\limits ^\{np\}\} A$, 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.},
author = {Bojan Magajna},
journal = {Studia Mathematica},
keywords = {operator space; von Neumann algebra; Haagerup tensor product; Pisier delta tensor product; completely bounded maps},
language = {eng},
number = {2},
pages = {137-164},
title = {On completely bounded bimodule maps over W*-algebras},
url = {http://eudml.org/doc/284437},
volume = {154},
year = {2003},
}

TY - JOUR
AU - Bojan Magajna
TI - On completely bounded bimodule maps over W*-algebras
JO - Studia Mathematica
PY - 2003
VL - 154
IS - 2
SP - 137
EP - 164
AB - 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 $I_{∞,∞}$. 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 $q:A {⊗\limits ^{eh}} X ⊗{\limits ^{eh}} A → X {⊗\limits ^{np}} A$, 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.
LA - eng
KW - operator space; von Neumann algebra; Haagerup tensor product; Pisier delta tensor product; completely bounded maps
UR - http://eudml.org/doc/284437
ER -