Third-power Associative Absolute Valued Algebras with a Nonzero Idempotent Commuting with All Idempotents
نویسندگان
چکیده
This paper deals with the determination of the absolute valued algebras with a nonzero idempotent commuting with the remaining idempotents and satisfying x2x = xx2 for every x. We prove that, in addition to the absolute valued algebras R, C, H, or O of the reals, complexes, division real quaternions or division real octonions, one such absolute valued algebra A can also be isometrically isomorphic to some of the absolute valued algebras ? C, ? H, or ? O, obtained from C, H, and O by imposing a new product defined by multiplying the conjugates of the elements. In particular, every absolute valued algebra having the above properties is finite-dimensional. This generalizes some well known theorems of Albert, Urbanik and Wright, and El-Mallah. 2010 Mathematics Subject Classification: 17A80, 17A75, 17A60, 17D99.
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