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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Weakly compact approximation in Banach spaces


Authors: Edward Odell and Hans-Olav Tylli
Journal: Trans. Amer. Math. Soc. 357 (2005), 1125-1159
MSC (2000): Primary 46B28; Secondary 46B25, 46B45
Published electronically: October 7, 2004
MathSciNet review: 2110435
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Abstract: The Banach space $E$ has the weakly compact approximation property (W.A.P. for short) if there is a constant $C < \infty $so that for any weakly compact set $D \subset E$ and $\varepsilon > 0$there is a weakly compact operator $V: E \to E$ satisfying $\sup _{x\in D} \Vert x - Vx \Vert < \varepsilon $ and $\Vert V\Vert \leq C$. We give several examples of Banach spaces both with and without this approximation property. Our main results demonstrate that the James-type spaces from a general class of quasi-reflexive spaces (which contains the classical James' space $J$) have the W.A.P, but that James' tree space $JT$ fails to have the W.A.P. It is also shown that the dual $J^{*}$ has the W.A.P. It follows that the Banach algebras $W(J)$ and $W(J^{*})$, consisting of the weakly compact operators, have bounded left approximate identities. Among the other results we obtain a concrete Banach space $Y$ so that $Y$ fails to have the W.A.P., but $Y$ has this approximation property without the uniform bound $C$.


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Additional Information

Edward Odell
Affiliation: Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712
Email: odell@math.utexas.edu

Hans-Olav Tylli
Affiliation: Department of Mathematics and Statistics, University of Helsinki, P.B. 68 (Gustaf Hällströmin katu 2b), FIN-00014 Finland
Email: hojtylli@cc.helsinki.fi

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03684-0
PII: S 0002-9947(04)03684-0
Received by editor(s): September 25, 2003
Published electronically: October 7, 2004
Additional Notes: The first author’s research was supported by the NSF
The second author’s research was supported by the Academy of Finland Project # 53893
Article copyright: © Copyright 2004 American Mathematical Society