Linear maps preserve the covariance sets

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$\ are $C^{\ast }$-algebras\textbf{.} A\textbf{ }linear map $\varphi :\mathcal{A}\to \mathcal{B}$ 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.

5 pages