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On absolute valued algebras with involution

✍ Scribed by MohamedLamei El-Mallah; Hader Elgendy; Abdellatif Rochdi; ÁngelRodríguez Palacios

Publisher
Elsevier Science
Year
2006
Tongue
English
Weight
133 KB
Volume
414
Category
Article
ISSN
0024-3795

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✦ Synopsis

Let A be an absolute valued algebra with involution, in the sense of Urbanik [K. Urbanik, Absolute valued algebras with an involution, Fund. Math. 49 (1961) 247-258]. We prove that A is finite-dimensional if and only if the algebra obtained by symmetrizing the product of A is simple, if and only if eA s = A s , where e denotes the unique nonzero self-adjoint idempotent of A, and A s stands for the set of all skew elements of A. We determine the idempotents of A, and show that A is the linear hull of the set of its idempotents if and only if A is equal to either McClay's algebra [A.A. Albert, A note of correction, Bull. Amer. Math. Soc. 55 (1949) 1191], the para-quaternion algebra, or the para-octonion algebra. We also prove that, if A is infinite-dimensional, then it can be enlarged to an absolute valued algebra with involution having a nonzero idempotent different from the unique nonzero self-adjoint idempotent.

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