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Weakly compact approximation in Banach spaces | Edward Odell
; Hans-Olav Tylli
; | Date: |
24 Sep 2003 | Subject: | Functional Analysis MSC-class: 46B28 | math.FA | 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 $epsilon > 0$ there is a weakly compact operator $V: E o E$ satisfying $sup_{xin D} || x - Vx || < epsilon$ and $|| V|| leq C$. 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 $C$. | Source: | arXiv, math.FA/0309405 | Services: | Forum | Review | PDF | Favorites |
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