Abstract Hermitian Algebras I. Spectral Resolution

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We refer to the real Jordan Banach algebra of bounded Hermitian operators on a Hilbert space as a Hermitian algebra. We define an abstract Hermitian algebra (AH-algebra) to be the directed group of an e-ring that contains a semitransparent element, has the quadratic annihilation property, and satisfies a Vigier condition on pairwise commuting ascending sequences. All of this terminology is explicated in this article, where we launch a study of AH-algebras. Here we establish the fundamental properties of AH-algebras, including the existence of polar decompositions and spectral resolutions, and we show that two elements of an AH-algebra commute if and only if their spectral projections commute. We employ spectral resolutions to assess the structure of maximal pairwise commuting subsets of an AH-algebra.
26 pages

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Functional Analysis, 06F25; 47B15

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http://salesiana.dossiersoluciones.com/handle/123456789/140874

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/0710.5062
 http://arxiv.org/abs/0710.5062

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arXiv