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STRONG SKEW COMMUTATIVITY PRESERVING MAPS ON RINGS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 100 / Issue 1 / February 2016
- Published online by Cambridge University Press:
- 25 November 2015, pp. 78-85
- Print publication:
- February 2016
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- Article
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- You have access
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Let
${\mathcal{A}}$ be a unital ring with involution. Assume that 
${\mathcal{A}}$ contains a nontrivial symmetric idempotent and 
${\it\phi}:{\mathcal{A}}\rightarrow {\mathcal{A}}$ is a nonlinear surjective map. We prove that if 
${\it\phi}$ preserves strong skew commutativity, then 
${\it\phi}(A)=ZA+f(A)$ for all 
$A\in {\mathcal{A}}$, where 
$Z\in {\mathcal{Z}}_{s}({\mathcal{A}})$ satisfies 
$Z^{2}=I$ and 
$f$ is a map from 
${\mathcal{A}}$ into 
${\mathcal{Z}}_{s}({\mathcal{A}})$. Related results concerning nonlinear strong skew commutativity preserving maps on von Neumann algebras are given.
