Transactions of the American Mathematical Society

Author(s): Eve Oja
Journal: Trans. Amer. Math. Soc. 352 (2000), 2801-2823.
MSC (2000): Primary 46B20, 46B28, 47L05
Posted: February 14, 2000
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Abstract: Let $a,c\geq 0$ and let

be a compact set of scalars. We introduce property $M^{\ast }(a,B,c)$ of Banach spaces

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

-, and

-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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Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46B20, 46B28, 47L05

Eve Oja
Affiliation: Faculty of Mathematics, Tartu University, Vanemuise 46, 51014 Tartu, Estonia
Email: eveoja@math.ut.ee

DOI: 10.1090/S0002-9947-00-02521-6
PII: S 0002-9947(00)02521-6
Received by editor(s): March 26, 1998
Posted: 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.
Copyright of article: Copyright 2000, American Mathematical Society

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