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THE ${\mathcal{K}}_{\mathit{up}}$ -APPROXIMATION PROPERTY AND ITS DUALITY

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

Published online by Cambridge University Press:  20 November 2014

JU MYUNG KIM
JU MYUNG KIM*
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul, 151-747, Korea email kjm21@kaist.ac.kr
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Abstract

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We introduce an approximation property ( ${\mathcal{K}}_{\mathit{up}}$ -AP, $1\leq p<\infty$ ), which is weaker than the classical approximation property, and discover the duality relationship between the ${\mathcal{K}}_{\mathit{up}}$ -AP and the ${\mathcal{K}}_{p}$ -AP. More precisely, we prove that for every $1<p<\infty$ , if the dual space $X^{\ast }$ of a Banach space $X$ has the ${\mathcal{K}}_{\mathit{up}}$ -AP, then $X$ has the ${\mathcal{K}}_{p}$ -AP, and if $X^{\ast }$ has the ${\mathcal{K}}_{p}$ -AP, then $X$ has the ${\mathcal{K}}_{\mathit{up}}$ -AP. As a consequence, it follows that every Banach space has the ${\mathcal{K}}_{u2}$ -AP and that for every $1<p<\infty$ , $p\neq 2$ , there exists a separable reflexive Banach space failing to have the ${\mathcal{K}}_{\mathit{up}}$ -AP.

Keywords

p-compact operatorunconditionally p-compact operatorKp -approximation propertyK up -approximation property

MSC classification

Primary: 46B28: Spaces of operators; tensor products; approximation properties
Secondary: 46B45: Banach sequence spaces 47L20: Operator ideals

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 98 , Issue 3 , June 2015 , pp. 364 - 374
Copyright
© 2014 Australian Mathematical Publishing Association Inc. 

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