Abstract
This paper deals with pairs of nonzero idempotents e and f of a third-power associative absolute valued algebra A satisfying \((ef)e= e(fe)\) and \((fe)f=f(ef)\) (pairwise flexible idempotents), and the role that they play on the structure of A. We show that if g is a nonzero idempotent of A such that the nonzero idempotents commuting with g are pairwise flexible, then the subalgebra that they generate \(B_g\) is isometrically isomorphic to \({\mathbb {R}}\), \(\mathop {{\mathbb {C}}}\limits ^{\star }\), \(\mathop {{\mathbb {H}}}\limits ^{\star }\), or \(\mathop {{\mathbb {O}}}\limits ^{\star }\). Our main theorem proves the equivalence of the following assertions: (i) for every two different nonzero idempotents e and f, the nonzero idempotents of A that commute with \((e-f)^2\) are pairwise flexible; (ii) each pair of nonzero idempotents of A generates a finite-dimensional subalgebra; and (iii) either A is isometrically isomorphic to \({\mathbb {R}}\), \({\mathbb {C}}\), \({\mathbb {H}}\), \({\mathbb {O}}\), \(\mathop {{\mathbb {C}}}\limits ^{\star }\), \(\mathop {{\mathbb {H}}}\limits ^{\star }\) or \(\mathop {{\mathbb {O}}}\limits ^{\star }\), or A contains a subalgebra B, which contains all idempotents of A and is isometrically isomorphic to the division absolute valued algebra \({\mathbb {P}}\) of the Okubo pseudo-octonions. More consequences on the structure of A related with the presence of pairwise flexible idempotents are given, among them several generalizations of some well-known theorems.
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This work was partially supported by Ministerio de Ciencia y Tecnologia (MTM2010-18370-C04-04 and MTM2013-45588-C3-C2).
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Communicated by V. Ravichandran.
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Cuenca Mira, J.A. On the Structure of Third-Power Associative Absolute Valued Algebras. Bull. Malays. Math. Sci. Soc. 40, 1135–1148 (2017). https://doi.org/10.1007/s40840-016-0351-1
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DOI: https://doi.org/10.1007/s40840-016-0351-1
Keywords
- Absolute valued algebra
- Idempotent
- Division algebra
- Third-power associativity
- Pairwise flexible idempotents