A projection–less approach to Rickart Jordan structures Garcés, Jorge J. Li, Lei Peralta Pereira, Antonio Miguel Tahlawi, Haifa A. Rickart C⁎-algebra, JB⁎-algebra and JB⁎-triple Baer C⁎-algebra and JB⁎-algebra Weakly order Rickart JB⁎-triple Von Neumann regularity, Inner ideal Acknowledgments We would like to express our gratitude to the anonymous referee for many constructive comments and suggestions to improve the final form of the paper. J. Garcés and A.M. Peralta partially supported by MCIN/AEI/10.13039/501-100011033/FEDER, EU, project no. PGC2018-093332-B-I00 and Junta de Andalucía grants number A-FQM-242-UGR18 and FQM375. L. Li partially supported by NSF of China (12171251) and Tianjin Natural Science Foundation (Grant No. 19JCY-BJC30200). A.M. Peralta is also supported by the IMAG–María de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033. H. Tahlawi supported by a grant of Scientific Research, King Saud University. Funding for open access charge: Universidad de Granada / CBUA. 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. 2022-09-13T11:15:35Z 2022-09-13T11:15:35Z 2022-06-22 info:eu-repo/semantics/article J.J. Garcés et al. A projection–less approach to Rickart Jordan structures. Journal of Algebra 609 (2022) 567–605 [https://doi.org/10.1016/j.jalgebra.2022.06.007] http://hdl.handle.net/10481/76680 10.1016/j.jalgebra.2022.06.007 eng http://creativecommons.org/licenses/by-nc-nd/4.0/ info:eu-repo/semantics/openAccess Attribution-NonCommercial-NoDerivatives 4.0 Internacional Elsevier