Brief communications

Preduals of von Neumann Algebras

A. I. Shtern

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: 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.

UDC: 517.98

Received: 18.11.2002

DOI: 10.4213/faa153

 English version: Functional Analysis and Its Applications, 2003, 37:2, 157–159

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