A projection–less approach to Rickart Jordan structures
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Rickart C⁎-algebra, JB⁎-algebra and JB⁎-triple Baer C⁎-algebra and JB⁎-algebraWeakly order Rickart JB⁎-tripleVon Neumann regularity, Inner ideal
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]
SponsorshipPartially supported by MCIN/AEI/10.13039/501-100011033/FEDER, EU, project no. PGC2018-093332-B-I00; Junta de Andalucía grants number A-FQM-242-UGR18 and FQM375; Partially supported by NSF of China (12171251); Tianjin Natural Science Foundation (Grant No. 19JCY-BJC30200); IMAG–María de Maeztu grant CEX2020-001105-M/AEI/10.13039/501100011033; 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.