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On the Radon-Nikodym property in Jordan algebras

Published online by Cambridge University Press:  18 May 2009

Cho-Ho Chu
Cho-Ho Chu
Affiliation:
University of London, Goldsmiths' College, London S.E.14
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Banach spaces whose duals possess the Radon-Nikodym property have been studied extensively in the past (cf. [5]). It has been shown recently in [4] that a C*-algebra is scattered if and only if its Banach dual possesses the Radon-Nikodym property. This result extends the well-known result of Pełczynski and Semandini [8] that a compact Hausdorff space Ωis dispersed if and only if C(Ω)* has the Radon-Nikodym property. The purpose of this note is to give a transparent proof of a more general result for Jordan algebras which unifies the aforementioned results. We prove that the dual of a JB-algebra A possesses the Radon-Nikodym property if and only if the state space of A is the cr-convex hull of its pure states. We also consider the projective tensor products of the duals of JB-algebras in this context.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 24 , Issue 2 , July 1983 , pp. 185 - 189
Copyright
Copyright © Glasgow Mathematical Journal Trust 1983

References

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