Geometric characterization of $L_1$-spaces

Abstract

The paper is devoted to a description of all strongly facially symmetric spaces which are isometrically isomorphic to $L_1$-spaces. We prove that if $Z$ is a real neutral strongly facially symmetric space such that every maximal geometric tripotent from the dual space of $Z$ is unitary then, the space $Z$ is isometrically isomorphic to the space $L_1(\Omega, \Sigma, \mu),$ where $(\Omega, \Sigma, \mu)$ is an appropriate measure space having the direct sum property.