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Nonlinear maps preserving the mixed product $_*[X\diamond Y, Z]$ on $*$-algebras

Raof Ahmad Bhat|Abbas Hussain Shikeh|Mohammad Aslam Siddeeque
대한수학회 (The Korean Mathematical Society)|2023.10|대한수학회논문집 (Communications of the KMS)|제38권 제4호|pp. 1019-1028|DOI 10.4134/CKMS.c230018

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참고문헌

키워드

isomorphism$*$-algebravon Neumann algebra
주제분류수학

초록

Let $\mathfrak{A}$ and $\mathfrak{B}$ be unital prime $*$-algebras such that $\mathfrak{A}$ contains a nontrivial projection. In the present paper, we show that if a bijective map $\Theta:\mathfrak{A}\to\mathfrak{B}$ satisfies $\Theta(_*[X\diamond Y, Z])={}_*[\Theta(X)\diamond \Theta(Y), \Theta(Z)]$ for all $X, Y, Z\in\mathfrak{A}$, then $\Theta$ or $-\Theta$ is a $*$-ring isomorphism. As an application, we shall characterize such maps in factor von Neumann algebras.

Abstract

Let $\mathfrak{A}$ and $\mathfrak{B}$ be unital prime $*$-algebras such that $\mathfrak{A}$ contains a nontrivial projection. In the present paper, we show that if a bijective map $\Theta:\mathfrak{A}\to\mathfrak{B}$ satisfies $\Theta(_*[X\diamond Y, Z])={}_*[\Theta(X)\diamond \Theta(Y), \Theta(Z)]$ for all $X, Y, Z\in\mathfrak{A}$, then $\Theta$ or $-\Theta$ is a $*$-ring isomorphism. As an application, we shall characterize such maps in factor von Neumann algebras.

참고문헌 (21)

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