Geometric and algebraic aspects of spectrality in order unit spaces: a comparison

TL;DR

This paper compares two spectral theories for order unit spaces, showing that Foulis's algebraic approach is more general than Alfsen and Shultz's geometric duality approach, with specific examples illustrating the differences.

Contribution

It demonstrates that Foulis's spectral compression bases encompass Alfsen-Shultz spectral duality, establishing a broader framework for spectral theory in order unit spaces.

Findings

Foulis approach is strictly more general than Alfsen-Shultz approach.

JB-algebras are Foulis spectral if and only if they are Rickart.

Centrally symmetric state spaces can be Foulis spectral without being Alfsen-Shultz spectral.

Abstract

Two approaches to spectral theory of order unit spaces are compared: the spectral duality of Alfsen and Shultz and the spectral compression bases due to Foulis. While the former approach uses the geometric properties of an order unit space in duality with a base norm space, the latter notion is purely algebraic. It is shown that the Foulis approach is strictly more general and contains the Alfsen-Shultz approach as a special case. This is demonstrated on two types of examples: the JB-algebras which are Foulis spectral if and only if they are Rickart, and the centrally symmetric state spaces, which may be Foulis spectral while not necessarily Alfsen-Shultz spectral.