Abstract Hermitian Algebras I. Spectral Resolution

TL;DR

This paper introduces the concept of abstract Hermitian algebras (AH-algebras), establishing their fundamental properties, including spectral resolutions and polar decompositions, and explores their structure and commutativity conditions.

Contribution

It defines AH-algebras and proves key properties like spectral resolutions and polar decompositions, advancing the understanding of Hermitian algebra structures.

Findings

Existence of spectral resolutions in AH-algebras

Commutativity characterized by spectral projections

Structure analysis of maximal commuting subsets

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.