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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Geometry of Banach spaces having shrinking approximations of the identity
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by Eve Oja PDF
Trans. Amer. Math. Soc. 352 (2000), 2801-2823 Request permission

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 }\| ax_{\nu }^{\ast } +bx^{\ast }+cy^{\ast }\|\leq \limsup _{\nu }\| x_{\nu }^{\ast }\|\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 $\| y^{\ast }\|\leq \| x^{\ast }\|$. Using $M^{\ast }(a,B,c)$ with $\max |B|+c>1$, 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
  • Email: eveoja@math.ut.ee
  • 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.
  • © Copyright 2000 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 352 (2000), 2801-2823
  • MSC (2000): Primary 46B20, 46B28, 47L05
  • DOI: https://doi.org/10.1090/S0002-9947-00-02521-6
  • MathSciNet review: 1675226