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
DOI:
https://doi.org/10.1090/S0002-9939-01-06142-1
Published electronically:
April 24, 2001
MathSciNet review:
1840107
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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
MR Author ID:
45945
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
MR Author ID:
95220
Email:
johnson@math.tamu.edu
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
