On absolute valued algebras with involution

• Published in: Linear algebra and its applications Vol. 414; no. 1; pp. 295 - 303
• Main Authors: El-Mallah, MohamedLamei, Elgendy, Hader, Rochdi, Abdellatif, Palacios, ÁngelRodríguez
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 01.04.2006; Elsevier Science
• 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; Involution; Normed space; Hilbert space; Absolute valued algebra; Algebra; Quaternion; Infinite dimension; Idempotent matrix; Self adjoint operator
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2005.10.005

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.