Nonlinear mixed type product $ [\mathscr{K}, \mathscr{F}]_\ast \odot \mathscr{D} $ on $ \ast $-algebras (2024)

Let $ \mathcal{A} $ be a unital $ \ast $-algebra containing a non-trivial projection. In this paper, we prove that if a map $ \Omega $ : $ \mathcal{A} $ $ \to $ $ \mathcal{A} $ such that

$ \begin{equation} \nonumber \Omega( [ \mathscr{K}, \mathscr{F}]_\ast \odot \mathscr{D}) = [\Omega(\mathscr{K}), \mathscr{F}]_\ast \odot \mathscr{D} + [ \mathscr{K}, \Omega (\mathscr{F})]_\ast \odot \mathscr{D} + [ \mathscr{K}, \mathscr{F}]_\ast \odot \Omega (\mathscr{D}), \end{equation} $

where $ [\mathscr{K}, \mathscr{F}]_{\ast} = \mathscr{K}\mathscr{F}- \mathscr{F}\mathscr{K}^\ast $ and $ \mathscr{K} \odot \mathscr{F} = \mathscr{K}^\ast \mathscr{F}+ \mathscr{F}\mathscr{K}^\ast $ for all $ \mathscr{K}, \mathscr{F}, \mathscr{D} \in \mathcal{A}, $ then $ \Omega $ is an additive $ \ast $-derivation. Furthermore, we extend its results on factor von Neumann algebras, standard operator algebras and prime $ \ast $-algebras. Additionally, we provide an example illustrating the existence of such maps.

    Keywords:
  • mixed bi-skew Jordan triple derivation,
  • $ \ast $-derivation,
  • $ \ast $- algebra,
  • involution

Citation: Junaid Nisar, Turki Alsuraiheed, Nadeem ur Rehman. Nonlinear mixed type product $ [\mathscr{K}, \mathscr{F}]_\ast \odot \mathscr{D} $ on $ \ast $-algebras[J]. AIMS Mathematics, 2024, 9(8): 21596-21608. doi: 10.3934/math.20241049

Abstract

Let $ \mathcal{A} $ be a unital $ \ast $-algebra containing a non-trivial projection. In this paper, we prove that if a map $ \Omega $ : $ \mathcal{A} $ $ \to $ $ \mathcal{A} $ such that

$ \begin{equation} \nonumber \Omega( [ \mathscr{K}, \mathscr{F}]_\ast \odot \mathscr{D}) = [\Omega(\mathscr{K}), \mathscr{F}]_\ast \odot \mathscr{D} + [ \mathscr{K}, \Omega (\mathscr{F})]_\ast \odot \mathscr{D} + [ \mathscr{K}, \mathscr{F}]_\ast \odot \Omega (\mathscr{D}), \end{equation} $

where $ [\mathscr{K}, \mathscr{F}]_{\ast} = \mathscr{K}\mathscr{F}- \mathscr{F}\mathscr{K}^\ast $ and $ \mathscr{K} \odot \mathscr{F} = \mathscr{K}^\ast \mathscr{F}+ \mathscr{F}\mathscr{K}^\ast $ for all $ \mathscr{K}, \mathscr{F}, \mathscr{D} \in \mathcal{A}, $ then $ \Omega $ is an additive $ \ast $-derivation. Furthermore, we extend its results on factor von Neumann algebras, standard operator algebras and prime $ \ast $-algebras. Additionally, we provide an example illustrating the existence of such maps.



Nonlinear mixed type product $ [\mathscr{K}, \mathscr{F}]_\ast \odot \mathscr{D} $ on $ \ast $-algebras (1)

References

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Nonlinear mixed type product $ [\mathscr{K}, \mathscr{F}]_\ast \odot \mathscr{D} $ on $ \ast $-algebras (2024)

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