Abstract
This paper deals with absolute valued algebras A satisfying \(x^2 x= xx^2\) for all \(x\in A\) and with each pair of different nonzero idempotents e and f of A being pairwise flexible in the sense that \((ef)e = e(fe)\) and \((fe)f = f(ef).\) We prove that any such absolute valued algebra A is isometrically isomorphic to \(\mathbb {R}\), \(\mathbb {C}\), \(\mathbb {H}\), \(\mathbb {O}\), \(\mathop {\mathbb {C}}\limits ^{\star }\), \(\mathop {\mathbb {H}}\limits ^{\star }\), \(\mathop {\mathbb {O}}\limits ^{\star }\), or \(\mathbb {P}.\) Several equivalences are given in addition in Theorem 3.2. This generalizes some well-known results of Albert, Urbanik and Wright, El–Mallah, and the author.
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This work was partially supported by Ministerio de Economía y Competitividad (MTM2013-45588-C3-2-P).
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Cuenca Mira, J.A. Third-Power Associative Absolute Valued Algebras with All Its Idempotents Pairwise Flexible. Mediterr. J. Math. 14, 196 (2017). https://doi.org/10.1007/s00009-017-0996-5
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DOI: https://doi.org/10.1007/s00009-017-0996-5