Description of Facially Symmetric Spaces with Unitary Tripotents

Abstract

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