Products of commuting Boolean algebras of projections and Banach space geometry

• Autores: Ben de Pagter , Werner J. Ricker
• Localización: Proceedings of the London Mathematical Society, ISSN 0024-6115, Vol. 91, Nº 2, 2005, págs. 483-508
• Idioma: inglés
• DOI: 10.1112/s0024611505015303
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• Resumen

  • 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 $-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.