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

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Abstract | References | Similar Articles | Additional Information

Following Davie's example of a Banach space failing the approximation property (1973), we show how to construct a Banach space which is asymptotically Hilbertian and fails the approximation property. Moreover, the space 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:
https://doi.org/10.1090/S0002-9939-01-06142-1

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