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

Raof Bhat; Abbas Shikeh; Mohammad Siddeeque

doi:10.4134/CKMS.c230018

Communications of the
Korean Mathematical Society CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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Commun. Korean Math. Soc. 2023; 38(4): 1019-1028

Online first article October 19, 2023 Printed October 31, 2023

https://doi.org/10.4134/CKMS.c230018

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

Raof Ahmad Bhat, Abbas Hussain Shikeh, Mohammad Aslam Siddeeque

Aligarh Muslim University; Aligarh Muslim University; Aligarh Muslim University

Abstract

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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.

Keywords: $*$-algebra, isomorphism, von Neumann algebra

MSC numbers: Primary 16W20, 47B48