Abstract
We study the absolute valued algebras containing a central element non necessary idempotent. We determine the absolute valued algebras containing a central element if we add some requirements. Also we gives a classification of finite-dimensional absolute valued algebras containing a generalized left unit and central element.
Highlights
The absolute valued algebras are introduced by Ostrowski in 1918
We study the absolute valued algebras containing a central element non necessary idempotent
If A is a finite dimensional absolute valued algebra, A has dimension 1, 2, 4 or 8 (Bott, et al, 1958; Ke(rvaire, 1958), A is isotopic to R, C, H or O and the norm of A comes from an inner product(Albert, 1947)
Summary
The absolute valued algebras are introduced by Ostrowski in 1918. It’s the normed algebra A such that ∥xy∥ = ∥x∥∥y∥ for all x, y in A. If A is a finite dimensional absolute valued algebra, A has dimension 1, 2, 4 or 8 (Bott, et al, 1958; Ke(rvaire, 1958), A is isotopic to R, C, H or O and the norm of A comes from an inner product(Albert, 1947). We have in Beslimane & Moutassim, 2011; Diankha, et al, 2013) a classification of absolute valued algebras with left unit and containing a central element. The norm of absolute valued algebra containing a central idempotent c, comes from to an inner product and the isometric map x → x⋆ := 2(x|c)c − x is an involution (El-Mallah, 1990). In this work we give a characterization of finite dimensional absolute valued algebra containing a central element. We determine the finite-dimensional absolute algebra containing a genaralized left unit and central element. We classify the absolute valued algebra containing a central element if we add some conditions
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