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Additivity of Jordan \(n\)-tuple maps on rings. (English) Zbl 1447.16039

Let \(n\geq2\). A pair \((M,M^{*})\) of maps \(M:R\to R'\) and \(M^{*}:R'\to R\) on rings \(R\) and \(R'\) ia a Jordan \(n\)-tuple maps of \(R\times R'\) if \[ M(M^{*}(x_{n})\circ(\dots(M^{*}(x_{2})\circ a_{1})\dots))=x_{n}\circ(\dots(x_{2}\circ M(a_{1}))\dots) \] and \[ M^{*}(M(a_{n})\circ(\dots(M(a_{2})\circ x_{1})\dots))=a_{n}\circ(\dots(a_{2}\circ M^{*}(x_{1}))\dots) \] for all \(a_{1},\dots,a_{n}\in R\) and \(x_{1},\dots,x_{n}\in R'\). When both \(M\) and \(M^{*}\) are additive (resp., injective, surjective, bijective), \((M,M^{*})\) is said to be additive (resp., injective, surjective, bijective).
The main result of this article states that if \(R\) is a \(2\)-torsion free ring which has a non-trivial idempontent \(e_{1}\) such that (1) \(e_{i}ce_{j}Re_{k}=0\) or \(e_{k}Re_{i}ce_{j}=0\) implies \(e_{i}ce_{j}=0\) \((1\leq i,j,k\leq2)\), and (2) \(e_{2}c_{1}e_{2}\circ(\dots(e_{2}c_{1}e_{2}\circ e_{2}ce_{2})\dots)=0\) for all \(c_{1}\in R\) implies \(e_{2}ce_{2}=0\), then every surjective Jordan \(n\)-tuple map \((M,M^{*})\) of \(R\times R'\) is additive.
As an application, if \(\mathfrak{X}\) is a Banach space with \(\dim\mathfrak{X}>1\), \(\mathfrak{A}\subseteq\mathfrak{B}(\mathfrak{X})\) a standard operator algebra and \(\mathfrak{A}'\) a ring, and if \(\phi:\mathfrak{A}\to\mathfrak{A}'\) is bijective map satisfying \[ \phi(a_{n}\circ(\dots(a_{2}\circ a_{1})\dots)=\phi(a_{n})\circ(\dots(\phi(a_{2})\circ\phi(a_{1}))\dots) \] for all \(a_{1},\dots,a_{n}\in\mathfrak{A}\), then \((\phi,\phi^{-1})\) is a surjective Jordan \(n\)-tuple map on \(\mathfrak{A}\times\mathfrak{A}'\), and so \(\phi\) is additive.

MSC:

16W10 Rings with involution; Lie, Jordan and other nonassociative structures
47B49 Transformers, preservers (linear operators on spaces of linear operators)
47L10 Algebras of operators on Banach spaces and other topological linear spaces
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