A projection--less approach to Rickart Jordan structures

TL;DR

This paper introduces and characterizes weakly Rickart JB$^*$-triples, establishing their properties and connections to C$^*$-algebras without relying on projections, thus advancing the understanding of Jordan structures.

Contribution

It defines weakly order Rickart JB$^*$-triples and proves their equivalence to weakly Rickart C$^*$-algebras, extending classical properties to this broader context.

Findings

Weakly order Rickart JB$^*$-triples are generated by their tripotents.

A C$^*$-algebra is a weakly Rickart JB$^*$-triple iff it is a weakly Rickart C$^*$-algebra.

Peirce-2 subspaces in these triples are Rickart JB$^*$-algebras.

Abstract

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