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

In this note we give a unified approach to the equivalence between the Krein-Milman property and the Radon-Nikodym property in Banach lattices and preduals of von Neumann algebras. It is by now well known that, for a Banach lattice, the Krein-Milman property (abbreviated KMP) and the Radon-Nikodym property (RNP) are equivalent [2]. The proof given in [2] combines techniques of [1] and [4]. On the other hand, as remarked in [3], KMP and RNP are also equivalent for preduals of von Neumann algebras. It is our purpose in this note to give a short proof of both results using the main result of [1] and an easy argument. We prove a more general result, in terms of ordered Banach spaces, enhancing both results above. Let E be an ordered Banach space [6] with a positive cone E+. Then the bidual E" of E is also an ordered Banach space whose positive cone will be denoted by E+. We identify E with the canoinical image of E into E". We say that E+ is solid in E+ if, for every z E E+ and every x E E such that z O, a + b = 1, and zl, Z2 E D such that z az1 + bz2. If zi E E, then bz2 E E. Since b > 0, Z2 E E. Hence, Z1, Z2 E D n E = D and z V ex(D). This contradiction proves that z1 V E. Similarly, Z2 E. But D C E", z1 < a-lz, Z2 < b-lz together with our assumption imply that Z1, z2 E E. This contradiction shows that ex(D) = 0 and C does not have the KMP. A Banach lattice E with the KMP does not contain a copy of co. Hence E is weakly sequentially complete and E+ is solid in E" [5, 11.5]. Taking C {x e E+: lixil < 1}, Theorem 1 proves that C is a Radon-Nikodym set. Therefore, positive linear operators from L1 [0, 1] into E are representable. Since every Received by the editors December 24, 1986. 1980 Mathemattics Sub5ject Classification (1985 Revision). Primary 46B22; Secondary 46B30.

A short proof of the equivalence of KMP and RNP in Banach lattices and preduals of von Neumann algebras