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

 

On intersections of ranges of projections of norm one in Banach spaces


Author: T. S. S. R. K. Rao
Journal: Proc. Amer. Math. Soc. 141 (2013), 3579-3586
MSC (2010): Primary 47L05; Secondary 46B20, 46E15
Published electronically: July 9, 2013
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Abstract: In this short note we are interested in studying Banach spaces in which the range of a projection of norm one whose kernel is of finite dimension is the intersection of ranges of finitely many projections of norm one whose kernels are of dimension one. We show that for a certain class of Banach spaces $ X$, the natural duality between $ X$ and $ X^{\ast \ast }$ can be exploited when the range of the projection is of finite codimension. We show that if $ X^\ast $ is isometric to $ L^1(\mu )$, then any central subspace of finite codimension is an intersection of central subspaces of codimension one. These results extend a recent result of Bandyopadhyay and Dutta which was proved for ranges of projections of norm one with finite dimensional kernel in continuous function spaces and unifies some earlier work of Baronti and Papini.


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Additional Information

T. S. S. R. K. Rao
Affiliation: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, R. V. College P. O., Bangalore 560059, India
Email: tss@isibang.ac.in

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11639-4
PII: S 0002-9939(2013)11639-4
Keywords: Projections of norm one, finite codimensional subspaces, spaces of continuous functions
Received by editor(s): July 11, 2011
Received by editor(s) in revised form: January 5, 2012
Published electronically: July 9, 2013
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.