Mappings preserving sum of triple products on $\ast $-algebras

TL;DR

This paper characterizes bijective maps between unital complex *-algebras that preserve a specific sum of triple products, revealing their structure under certain scalar conditions, with applications demonstrated.

Contribution

It provides a detailed structure theorem for mappings preserving a sum of triple products in *-algebras, extending understanding of algebraic preservers.

Findings

Characterization of structure of bijective preservers

Conditions on scalars for preservation

Applications to algebraic mappings

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be two unital complex $\ast $-algebras such that $\mathcal{A}$ has a nontrivial projection. In this paper, we study the structure of bijective mappings $\Phi :\mathcal{A}\rightarrow \mathcal{B}$ preserving sum of triple products $\alpha_{1} ab^{*}c+\alpha_{2} acb^{*}+\alpha_{3} b^{*}ac +\alpha_{4} cab^{*}+\alpha_{5} b^{*}ca+\alpha_{6} cb^{*}a,$ where the scalars $\{\alpha_{k}\}_{k=1}^{6}$ are complex numbers satisfying some conditions. Applications of obtained results are given.