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

   

 

An example of an asymptotically Hilbertian space which fails the approximation property


Authors: P. G. Casazza, C. L. García and W. B. Johnson
Journal: Proc. Amer. Math. Soc. 129 (2001), 3017-3023
MSC (2000): Primary 46B20, 46B07, 46B28; Secondary 46B99
Published electronically: April 24, 2001
MathSciNet review: 1840107
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Abstract:

Following Davie's example of a Banach space failing the approximation property (1973), we show how to construct a Banach space $E$ which is asymptotically Hilbertian and fails the approximation property. Moreover, the space $E$ is shown to be a subspace of a space with an unconditional basis which is ``almost'' a weak Hilbert space and which can be written as the direct sum of two subspaces all of whose subspaces have the approximation property.


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Additional Information

P. G. Casazza
Affiliation: Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211
Email: pete@math.missouri.edu

C. L. García
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843–3368
Email: clgarcia@math.tamu.edu

W. B. Johnson
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843–3368
Email: johnson@math.tamu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06142-1
PII: S 0002-9939(01)06142-1
Keywords: Banach spaces, weak Hilbert spaces, asymptotically Hilbertian, approximation property
Received by editor(s): March 1, 2000
Published electronically: April 24, 2001
Additional Notes: The first author was supported by NSF grant DMS-970618.
The second and third authors were supported in part by NSF grants DMS-9623260, DMS-9900185, and by the Texas Advanced Research Program under Grant No. 010366-163.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2001 P. G. Casazza, C. L. García, and W. B. Johnson