Additivity of maps preserving Jordan $\eta_{\ast}$-products on $C^{*}$-algebras

TL;DR

This paper proves that bijective unital maps preserving Jordan st-products on prime C*-algebras are additive, and st-additive when ta is rational, expanding understanding of structure-preserving maps in operator algebras.

Contribution

It establishes the additivity and st-additivity of maps preserving Jordan st-products under specific conditions, a novel result in C*-algebra theory.

Findings

ta st-preserving maps are additive when |ta| 1.

Such maps are st-additive if ta is rational.

The results apply to prime C*-algebras with nontrivial projections.

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be two $C^{*}$-algebras such that $\mathcal{B}$ is prime. In this paper, we investigate the additivity of map $\Phi$ from $\mathcal{A}$ onto $\mathcal{B}$ that are bijective unital and satisfies $$\Phi(AP+\eta PA^{*})=\Phi(A)\Phi(P)+\eta \Phi(P)\Phi(A)^{*},$$ for all $A\in\mathcal{A}$ and $P\in\{P_{1},I_{\mathcal{A}}-P_{1}\}$ where $P_{1}$ is a nontrivial projection in $\mathcal{A}$. Let $\eta$ be a non-zero complex number such that $|\eta|\neq1$, then $\Phi$ is additive. Moreover, if $\eta$ is rational then $\Phi$ is $\ast$-additive.