Nonlinear maps preserving sums of triple products on $\ast $-algebras

TL;DR

This paper characterizes bijective nonlinear maps between unital complex *-algebras that preserve specific sums of triple products, revealing their structure under certain conditions on the scalars and the algebra's projections.

Contribution

It provides a detailed description of the structure of nonlinear sum-preserving maps on *-algebras, extending understanding of algebraic automorphisms and preservers.

Findings

Characterization of nonlinear sum-preserving maps

Conditions under which these maps are algebra automorphisms

Extension of known preserver results to more general triple products

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 nonlinear maps $\Phi :\mathcal{A}\rightarrow \mathcal{B}$ preserving sum of triple products $\alpha _{1} abc+\alpha _{2} a^{*}cb^{*}+\alpha _{3} ba^{*}c +\alpha _{4} cab^{*}+\alpha _{5} bca+\alpha _{6} cb^{*}a^{*},$ where the scalars $\{\alpha _{k}\}_{k=1}^{6}$ are complex numbers satisfying some conditions.