Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-06-27T18:09:31.105Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)