A projection--less approach to Rickart Jordan structures

Abstract

The main goal of this paper is to introduce and explore an appropriate notion of weakly Rickart JB${}^{\ast }$-triples. We introduce weakly order Rickart JB${}^{\ast }$-triples, and we show that a C${}^{\ast }$-algebra $A$ is a weakly (order) Rickart JB${}^{\ast }$-triple precisely when it is a weakly Rickart C${}^{\ast }$-algebra. We also prove that the Peirce-2 subspace associated with a tripotent in a weakly order Rickart JB${}^{\ast }$-triple is a Rickart JB${}^{\ast }$-algebra in the sense of Ayupov and Arzikulov. By extending a classical property of Rickart C${}^{\ast }$-algebras, we prove that every weakly order Rickart JB${}^{\ast }$-triple is generated by its tripotents.