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Geometry of Banach spaces having shrinking approximations of the identity

Author: Eve Oja
Journal: Trans. Amer. Math. Soc. 352 (2000), 2801-2823
MSC (2000): Primary 46B20, 46B28, 47L05
Published electronically: February 14, 2000
MathSciNet review: 1675226
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Abstract: Let $a,c\geq 0$ and let $B$ be a compact set of scalars. We introduce property $M^{\ast }(a,B,c)$ of Banach spaces $X$ by the requirement that \begin{equation*}\limsup _{\nu }\Vert ax_{\nu }^{\ast } +bx^{\ast }+cy^{\ast }\V... ...q \limsup _{\nu }\Vert x_{\nu }^{\ast }\Vert\quad \forall b\in B \end{equation*}whenever $(x_{\nu }^{\ast })$ is a bounded net converging weak$^{\ast }$ to $x^{\ast }$ in $X^{\ast }$ and $\Vert y^{\ast }\Vert\leq \Vert x^{\ast }\Vert$. Using $M^{\ast }(a,B,c)$ with $\max \vert B\vert+c>1$, we characterize the existence of certain shrinking approximations of the identity (in particular, those related to $M$-, $u$-, and $h$-ideals of compact or approximable operators). We also show that the existence of these approximations of the identity is separably determined.

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Additional Information

Eve Oja
Affiliation: Faculty of Mathematics, Tartu University, Vanemuise 46, 51014 Tartu, Estonia

Received by editor(s): March 26, 1998
Published electronically: February 14, 2000
Additional Notes: This work was completed during a visit of the author to Freie Universität Berlin in 1997, supported by a grant from the Deutscher Akademischer Austauschdienst. The research was also partially supported by Estonian Science Foundation Grants 1820 and 3055.
The author is grateful to D. Werner for his hospitality and useful conversations.
Article copyright: © Copyright 2000 American Mathematical Society