On absolute valued algebras with involution

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...

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Bibliographic Details
Published in:Linear algebra and its applications 2006-04, Vol.414 (1), p.295-303
Main Authors: El-Mallah, MohamedLamei, Elgendy, Hader, Rochdi, Abdellatif, Palacios, ÁngelRodríguez
Format: Article
Language:English
Subjects:
Absolute valued algebra
Algebra
Exact sciences and technology
Hilbert space
Involution
Linear and multilinear algebra, matrix theory
Mathematics
Normed space
Sciences and techniques of general use
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Summary: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.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2005.10.005