Ricerche di Matematica ( IF 0.970 ) Pub Date : 2020-11-20 , DOI: 10.1007/s11587-020-00547-z
Bruno Leonardo Macedo Ferreira, Ivan Kaygorodov

Suppose $${\mathfrak R}\,$$ is a 2,3-torsion free unital alternative ring having an idempotent element $$e_1$$ $$\left( e_2 = 1-e_1\right)$$ which satisfies $$x {\mathfrak R}\, \cdot e_i = \{0\} \Rightarrow x = 0$$ $$\left( i = 1,2\right)$$. In this paper, we aim to characterize the commuting maps. Let $$\varphi$$ be a commuting map of $${\mathfrak R}\,$$ so it is shown that $$\varphi (x) = zx + \Xi (x)$$ for all $$x \in {\mathfrak R}\,$$, where $$z \in \mathcal {Z}({\mathfrak R}\, )$$ and $$\Xi$$ is an additive map from $${\mathfrak R}\,$$ into $$\mathcal {Z}({\mathfrak R}\, )$$. As a consequence a characterization of anti-commuting maps is obtained and we provide as an application, a characterization of commuting maps on von Neumann algebras relative alternative $$C^{*}$$-algebra with no central summands of type $$I_1$$.

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