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THE QUOTIENT ALGEBRA OF COMPACT-BY-APPROXIMABLE OPERATORS ON BANACH SPACES FAILING THE APPROXIMATION PROPERTY

Part of: Normed linear spaces and Banach spaces; Banach lattices Linear spaces and algebras of operators

Published online by Cambridge University Press:  07 June 2019

HANS-OLAV TYLLI  and
HENRIK WIRZENIUS
HANS-OLAV TYLLI*
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, Box 68, Pietari Kalmin katu 5, FI-00014Helsinki, Finland
HENRIK WIRZENIUS
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, Box 68, Pietari Kalmin katu 5, FI-00014Helsinki, Finland e-mail: henrik.wirzenius@helsinki.fi
*
e-mail: hans-olav.tylli@helsinki.fi
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Abstract

We initiate a study of structural properties of the quotient algebra ${\mathcal{K}}(X)/{\mathcal{A}}(X)$ of the compact-by-approximable operators on Banach spaces $X$ failing the approximation property. Our main results and examples include the following: (i) there is a linear isomorphic embedding from $c_{0}$ into ${\mathcal{K}}(Z)/{\mathcal{A}}(Z)$, where $Z$ belongs to the class of Banach spaces constructed by Willis that have the metric compact approximation property but fail the approximation property, (ii) there is a linear isomorphic embedding from a nonseparable space $c_{0}(\unicode[STIX]{x1D6E4})$ into ${\mathcal{K}}(Z_{FJ})/{\mathcal{A}}(Z_{FJ})$, where $Z_{FJ}$ is a universal compact factorisation space arising from the work of Johnson and Figiel.

Keywords

quotient algebracompact-by-approximable operatorsapproximation properties

MSC classification

Primary: 46B28: Spaces of operators; tensor products; approximation properties 47L10: Algebras of operators on Banach spaces and other topological linear spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 110 , Issue 2 , April 2021 , pp. 266 - 288
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by A. Sims

H. Wirzenius gratefully acknowledges the financial support of The Swedish Cultural Foundation in Finland and the Magnus Ehrnrooth Foundation.

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