English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

A a-finite von Neumann algebra is a second dual if and only if it is atomic. Shtern [7] has shown that a C*-algebra is a third dual if and only if it is the enveloping von Neumann algebra of a von Neumann algebra. C*-algebras that are dual spaces are the von Neumann algebras. The remaining question appears to be: when is a C*-algebra a second dual? This question is the same as asking when a von Neumann algebra is a second dual. Let M be a von Neumann algebra and let M = X** for some Banach space X. By the Krein-Milman Theorem, M has a pure normal state and hence M has a type I summand. Can M have type II and type III summands as well? The answer is yes if, for instance, X is the quotient of the algebra B(H) of bounded operators on a separable Hilbert space H by the compact operators K(H) [1, 6]. The above observation seems to suggest, without some restriction, that the question may be too general to have a fruitful answer. If, however, we restrict the question to the class of a-finite von Neumann algebras, then a simple answer emerges. Theorem. A cr-finite von Neumann algebra M is a second dual space if and only if M is atomic, that is, if and only if M is a direct sum of the B(H)'s. In the commutative case, the above is of course a well-known result of Pelczynski [4, Theorem 4] since a commutative a-finite von Neumann algebra is just an LJ,(u) where ,u is a a-finite measure and in this case, Pelczynski has shown that L1 (y) is (isomorphic to) a dual if and only if ,u is purely atomic. Rosenthal [5, Corollary 2.2] has given a short proof of this result. Distel and Uhl [3, p. 83] gave a shorter proof using the Radon-Nikodym property and their approach works well in the noncommutative case. Recall that the a-finite von Neumann algebras are those that admit faithful normal states. Examples: (i) any von Neumann algebra acting on a separable Hilbert space; (ii) all von Neumann algebras appeared in quantum statistical mechanics and quantum field theory. A Banach space is called weakly compactly generated (WCG) if it is the (norm) closed linear span of one of its Received by the editors February 1, 1990 and, in revised form, August 20, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 46L10, 46B22. 01991 American Mathematical Society 0002-9939/91 $1.00 + $.25 per page

von Neumann algebras which are second dual spaces