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Products of commuting Boolean algebras of projections and Banach space geometry

Published online by Cambridge University Press:  23 August 2005

Ben de Pagter  and
Werner J. Ricker
Ben de Pagter
Affiliation:
Department of Applied Mathematics, Faculty EEMCS, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands. E-mail: b.depagter@ewi.tudelft.nl
Werner J. Ricker
Affiliation:
Math.-Geogr.Fakultät, Katholische Universität Eichstätt-Ingolstadt, D–85072 Eichstätt, Germany. E-mail: werner.ricker@ku-eichstaett.de
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Abstract

New criteria and Banach spaces are presented (for example, $GL$-spaces and Banach spaces with property $(\alpha)$) that ensure that the Boolean algebra generated by a pair of bounded, commuting Boolean algebras of projections is itself bounded. The notion of $R$-boundedness plays a fundamental role. It is shown that the strong operator closure of any $R$-bounded Boolean algebra of projections is necessarily Bade complete. Also, for a Dedekind $\sigma <formula form="inline" disc="math" id="frm006"><formtex notation="AMSTeX">$-complete Banach lattice $E$, the Boolean algebra consisting of all band projections in $E$ is $R$-bounded if and only if $E$ has finite cotype. In this situation, every bounded Boolean algebra of projections in $E$ is $R$-bounded and has a Bade complete strong closure.

Keywords

R-boundednessproduct Boolean algebrasproperty (α)GL-spacesband projections46B2047L10 (primary)46B4247B4047B60 (secondary)
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 91 , Issue 2 , September 2005 , pp. 483 - 508
Copyright
2005 London Mathematical Society

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