Spectral order unit spaces and JB-algebras

TL;DR

This paper explores the structure of spectral order unit spaces and JB-algebras, introducing functional calculi and characterizations that extend known results from C*-algebras to JB-algebras.

Contribution

It defines functional calculus for spectral order unit spaces and characterizes JB-algebras with the comparability property, extending existing theorems to a broader algebraic context.

Findings

Defined continuous and Borel functional calculus for spectral order unit spaces.

Characterized order unit spaces with the comparability property as JB-algebras.

Extended Saito and Wright's results to Rickart JB-algebras with monotone σ-complete maximal associative subalgebras.

Abstract

Order unit spaces with comparability and spectrality properties as introduced by Foulis are studied. We define continuous functional calculus for order unit spaces with the comparability property and Borel functional calculus for spectral order unit spaces. Applying the conditions of Alfsen and Schultz, we characterize order unit spaces with comparability property that are JB-algebras. We also prove a characterization of Rickart JB-algebras as those JB-algebras for which every maximal associative subalgebra is monotone $\sigma$-complete, extending an analogous result of Sait\^o and Wright for C*-algebras.