# Bulletin of theKorean Mathematical SocietyBKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

## Article

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Bull. Korean Math. Soc. 2019; 56(2): 451-459

Published online March 1, 2019 https://doi.org/10.4134/BKMS.b180327

## Maps preserving Jordan and $\ast$-Jordan triple product on operator $\ast$-algebras

Vahid Darvish, Mojtaba Nouri, Mehran Razeghi, Ali Taghavi

University of Mazandaran; University of Mazandaran; University of Mazandaran; University of Mazandaran

### Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be two operator $\ast$-rings such that $\mathcal{A}$ is prime. In this paper, we show that if the map $\Phi:\mathcal{A}\to\mathcal{B}$ is bijective and preserves Jordan or $\ast$-Jordan triple product, then it is additive. Moreover, if $\Phi$ preserves Jordan triple product, we prove the multiplicativity or anti-multiplicativity of $\Phi$. Finally, we show that if $\mathcal{A}$ and $\mathcal{B}$ are two prime operator $\ast$-algebras, $\Psi:\mathcal{A}\to\mathcal{B}$ is bijective and preserves $\ast$-Jordan triple product, then $\Psi$ is a $\mathbb{C}$-linear or conjugate $\mathbb{C}$-linear $\ast$-isomorphism.

Keywords: $\ast$-Jordan triple product, $\ast$-algebra

MSC numbers: 47B48, 46L10

Supported by: This research work has been supported by a research grant from the University of Mazandaran.