Abstract
The main goal of this paper is to introduce and explore an appropriate notion of weakly Rickart JB⁎-triples. We introduce weakly and 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 any 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.