Description of facially symmetric spaces with unitary tripotents

TL;DR

This paper characterizes finite-dimensional real neutral strongly facially symmetric spaces with property JP and shows that spaces with unitary tripotents are isometrically isomorphic to certain measure spaces, enriching the understanding of facially symmetric spaces.

Contribution

It provides a detailed description of facially symmetric spaces with property JP and establishes an isometric isomorphism to measure spaces for spaces with unitary tripotents.

Findings

Spaces with property JP are characterized.

Spaces with unitary tripotents are isometrically isomorphic to measure spaces.

Finite-dimensional real neutral strongly facially symmetric spaces are described explicitly.

Abstract

We give a description of finite-dimensional real neutral strongly facially symmetric spaces with the property JP. We also prove that if \(Z\) is a real neutral strongly facially symmetric with an unitary tripotents, then the space \(Z\) is isometrically isomorphic to the space \(L_1(\Omega,\Sigma, \mu),\) where \((\Omega,\Sigma, \mu)\) is a measure space having the direct sum property.