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Jordan triple homomorphisms on \(\mathcal{T}_{\infty} (F)\). (English) Zbl 1497.15004

Summary: Let \(\mathcal{T}_{\infty} (F)\) be the algebra of all \(\mathbb{N}\times\mathbb{N}\) upper triangular matrices defined over a field \(F\) of characteristic different from 2. We consider the Jordan triple homomorphisms of \(\mathcal{T}_{\infty} (F)\), i.e. the additive maps that satisfy the condition \(\phi (xyx)=\phi (x)\phi (y)\phi (x)\) for all \(x,y\in\mathcal{T}_{\infty} (F)\). For the case when \(F\) is a prime field we find the form of all such maps \(\phi\). For the general case we present the form of the surjective maps \(\phi\).

MSC:

15A04 Linear transformations, semilinear transformations
15A30 Algebraic systems of matrices
15B33 Matrices over special rings (quaternions, finite fields, etc.)
47B49 Transformers, preservers (linear operators on spaces of linear operators)
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