Linear maps preserve the covariance sets

TL;DR

This paper investigates how linear maps called $C^{ ext{*}}$-Jordan homomorphisms preserve covariance sets and cosets within $C^{ ext{*}}$-algebras, revealing structural preservation under mild conditions.

Contribution

It demonstrates that $C^{ ext{*}}$-Jordan homomorphisms preserve covariance sets and cosets in $C^{ ext{*}}$-algebras under mild assumptions, extending understanding of algebraic structure preservation.

Findings

Preservation of covariance sets by $C^{ ext{*}}$-Jordan homomorphisms.

Preservation of covariance cosets in $C^{ ext{*}}$-algebras.

Results hold under mild algebraic assumptions.

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$\ are $C^{{\huge \ast}}% $-algebras\textbf{.} A\textbf{ }linear map $\phi:\mathcal{A\rightarrow B}$ is $C^{\ast}$-Jordan homomorphism if it is a Jordan homomorphism which preserves the adjoint operation. In this note we show that $C^{\ast}$-Jordan homomorphisms -- under mild assumptions -- preserving covariance set and covariance coset in $C^{\ast}$-algebras.