Preduals of von Neumann Algebras
Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 2, pp. 92-94
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Proofs of two assertions are sketched. 1) If the Banach space of a von Neumann algebra $\mathfrak A$ is the third dual of some Banach space, then the space $\mathfrak A$ is isometrically isomorphic to the second dual of some von Neumann algebra $A$ and the von Neumann algebra $A$ is uniquely determined by its enveloping von Neumann algebra (up to von Neumann algebra isomorphism) and is the unique second predual of $\mathfrak A$ (up to isometric isomorphism of Banach spaces). 2) An infinite-dimensional von Neumann algebra cannot have
preduals of all orders.
Keywords:
von Neumann algebra, Banach space, dual, predual.
@article{FAA_2003_37_2_a10,
author = {A. I. Shtern},
title = {Preduals of von {Neumann} {Algebras}},
journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
pages = {92--94},
publisher = {mathdoc},
volume = {37},
number = {2},
year = {2003},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/FAA_2003_37_2_a10/}
}
A. I. Shtern. Preduals of von Neumann Algebras. Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 2, pp. 92-94. http://geodesic.mathdoc.fr/item/FAA_2003_37_2_a10/