On absolute valued algebras with involution |
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Authors: | MohamedLamei El-Mallah Hader Elgendy |
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Affiliation: | a DepartmentofMathematics,FacultyofScience,CairoUniversity,12613Giza,Egypt b DepartmentofMathematics,DamiettaFacultyofScience,NewDamiettaCity,34517Damietta,Egypt c DépartementdeMathématiquesetInformatique,FacultédesSciences,UniversitéHassanII,B.P.7955, Casablanca, Morocco d Facultad de Ciencias, Departamento de Análisis Matemático, Universidad de Granada, 18071 Granada, Spain |
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Abstract: | 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 eAs = As, where e denotes the unique nonzero self-adjoint idempotent of A, and As 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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Keywords: | Absolute valued algebra Involution Normed space Hilbert space |
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