Maps preserving strong Jordan multiple $*$-product on $*$-algebras

Document Type : Research Paper

Author

Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, P. O. Box 47416-1468, Babolsar, Iran

Abstract

Let $\mathcal{A}$ be an arbitrary $*$-algebra with unit I over the real or complex field $\mathbb{F} $ that contains a nontrivial idempotent $P_{1}$ and $ n\geq 1$ a natural number and $\varphi:\mathcal{A} \longrightarrow \mathcal{A}$ be a surjective map on $\mathcal{A}$ such that $\varphi$ satisfies condition
$$\varphi(P)\bullet_{n-1}\varphi(P)\bullet\varphi(A)=P\bullet_{n-1} P\bullet A,$$
  for every $ A \in\mathcal{A}$ and projection $P\in\{P_{1},~I-P_{1}\}$, where $ A\bullet_{n-1} A$ with repeat $ n-1 $ times $ A $ is the Jordan multiple $*$-product.   Then $\varphi(A)=\varphi(I)A$ for all $ A\in\mathcal{A}$ and $\varphi(I)^2=I$.

Keywords

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Volume 13, Issue 2
July 2022
Pages 221-225
  • Receive Date: 07 January 2020
  • Revise Date: 28 August 2021
  • Accept Date: 30 August 2021