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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Grothendieck ordered Banach spaces with an interpolation property


Authors: Ioannis A. Polyrakis and Foivos Xanthos
Journal: Proc. Amer. Math. Soc. 141 (2013), 1651-1661
MSC (2010): Primary 46B40, 46B42, 47B60
Published electronically: October 26, 2012
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Abstract: In this paper we prove that if $ E$ is an ordered Banach space with the countable interpolation property, $ E$ has an order unit and $ E_+$ is closed and normal, then $ E$ is a Grothendieck space; i.e. any weak-star convergent sequence of $ E^*$ is weakly convergent. By the countable interpolation property we mean that for any $ A,B\subseteq E$ countable, with $ A\leq B$, we have $ A\leq \{x\}\leq B$ for some $ x\in E$.


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Additional Information

Ioannis A. Polyrakis
Affiliation: Department of Mathematics, National Technical University of Athens, Zographou 157 80, Athens, Greece
Email: ypoly@math.ntua.gr

Foivos Xanthos
Affiliation: Department of Mathematics, National Technical University of Athens, Zographou 157 80, Athens, Greece
Email: fxanthos@math.ntua.gr

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11437-6
PII: S 0002-9939(2012)11437-6
Keywords: Ordered Banach spaces, Banach lattices, Grothendieck spaces, regular operators
Received by editor(s): February 14, 2011
Received by editor(s) in revised form: September 5, 2011
Published electronically: October 26, 2012
Additional Notes: This research was supported by the HERAKLEITOS II project, which is co-funded by the European Social Fund and National Resources
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.