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arXiv · 0710.5062

Abstract Hermitian Algebras I. Spectral Resolution

David J. Foulis·Sylvia Pulmannova
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Abstract

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.

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David J. Foulis, Sylvia Pulmannova. 2007-10-26. Abstract Hermitian Algebras I. Spectral Resolution. https://arxiv.org/abs/0710.5062

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