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Absolute valued algebras with involution. (English) Zbl 1177.46039

Absolute valued algebras with involution (avawi, for short), as defined in K.Urbanik’s paper [Fundam.Math.49, 247–258 (1961; Zbl 0267.17005)], are dealt with in this paper. The authors prove that any avawi satisfying the identity \((x, x^2, x)=0\), with \((\cdot, \cdot, \cdot)\) denoting the associator, is finite-dimensional, and that in dimension different from two, isomorphisms between avawi’s are in fact \(*\)-isomorphisms. They also classify finite-dimensional avawi’s and prove that there exists an eight-dimensional avawi which contains a nonzero central idempotent and where the group of its automorphisms is trivial.

MSC:

46K70 Nonassociative topological algebras with an involution
46H70 Nonassociative topological algebras
46E15 Banach spaces of continuous, differentiable or analytic functions
46B04 Isometric theory of Banach spaces

Citations:

Zbl 0267.17005
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References:

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