AL-Spaces and AM-Spaces of Operators

AL-Spaces and AM-Spaces of Operators Positivity 4: 303–311, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands. A.W. WICKSTEAD Department of Pure Mathematics, The Queen’s University of Belfast, Belfast, BT7 1NN, Northern Ireland 1. Introduction In the study of Banach lattices two special classes have long been the subject of special attention, namely AL-space and AM-spaces. Apart from the sheer beauty of Kakutani’s concrete characterisations of these spaces ([5] and [6]) they continue to play an important role in the current study of Banach lattices X because of the use of unital AM- spaces in their local study and of AL-spaces as the completions of quotients of the form X=fx V f.jxj/ D 0g for f 2 X . In spite of the, by now, considerable literature on spaces of regular operators between two Banach lattices, there seems to have been no discussion of when these spaces are either AL-spaces or AM-spaces although relevant results have been obtained in a rather more general setting by A.J. Ellis [4] and the author [9]. In this note we seek to remedy this oversight. Many of our results are to be expected although the role of the Fatou property in the case of AM-spaces of operators http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

AL-Spaces and AM-Spaces of Operators

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Publisher
Springer Journals
Copyright
Copyright © 2000 by Kluwer Academic Publishers
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1009803007323
Publisher site
See Article on Publisher Site

Abstract

Positivity 4: 303–311, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands. A.W. WICKSTEAD Department of Pure Mathematics, The Queen’s University of Belfast, Belfast, BT7 1NN, Northern Ireland 1. Introduction In the study of Banach lattices two special classes have long been the subject of special attention, namely AL-space and AM-spaces. Apart from the sheer beauty of Kakutani’s concrete characterisations of these spaces ([5] and [6]) they continue to play an important role in the current study of Banach lattices X because of the use of unital AM- spaces in their local study and of AL-spaces as the completions of quotients of the form X=fx V f.jxj/ D 0g for f 2 X . In spite of the, by now, considerable literature on spaces of regular operators between two Banach lattices, there seems to have been no discussion of when these spaces are either AL-spaces or AM-spaces although relevant results have been obtained in a rather more general setting by A.J. Ellis [4] and the author [9]. In this note we seek to remedy this oversight. Many of our results are to be expected although the role of the Fatou property in the case of AM-spaces of operators

Journal

PositivitySpringer Journals

Published: Oct 16, 2004

References

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