A Loomis-Sikorski theorem and functional calculus for a generalized Hermitian algebra

TL;DR

This paper extends the Loomis-Sikorski theorem to generalized Hermitian algebras, establishing a functional calculus and spectral theory for these structures, which generalize Hermitian operators on Hilbert spaces.

Contribution

It introduces the concept of gh-tribes and proves that every commutative GH-algebra is representable via these, enabling a spectral and functional calculus for GH-algebras.

Findings

Every commutative GH-algebra is the image of a gh-tribe.

Each element corresponds to a real observable with a spectral resolution.

A continuous functional calculus is established for GH-algebras.

Abstract

A generalized Hermitian (GH-) algebra is a generalization of the partially ordered Jordan algebra of all Hermitian operators on a Hilbert space. We introduce the notion of a gh-tribe, which is a commutative GH-algebra of functions on a nonempty set $X$ with pointwise partial order and operations, and we prove that every commutative GH-algebra is the image of a gh-tribe under a surjective GH-morphism. Using this result, we prove each element $a$ of a GH-algebra $A$ corresponds to a real observable $\xi_a$ on the $\sigma$-orthomodular lattice of projections in $A$ and that $\xi_a$ determines the spectral resolution of $a$. Also, if $f$ is a continuous function defined on the spectrum of $a$, we formulate a definition of $f(a)$, thus obtaining a continuous functional calculus for $A$.