Jordan $\ast$-homomorphisms on the spaces of continuous maps taking values in $C^{\ast }$-algebras

Abstract

Let $\mathcal{A}$ be a unital $C^{\ast }$-algebra. We consider Jordan $\ast$-homomorphisms on $C(X,\mathcal{A})$ and Jordan $\ast$-homomorphisms on $\mathrm{Lip}(X,\mathcal{A})$. More precisely, for any unital $C^{\ast }$-algebra $\mathcal{A}$, we prove that every Jordan $\ast$-homomorphism on $C(X,\mathcal{A})$ and every Jordan $\ast$-homomorphism on $\mathrm{Lip}(X,\mathcal{A})$ is represented as a weighted composition operator by using the irreducible representations of $\mathcal{A}$. In addition, when ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ are primitive $C^{\ast }$-algebras, we characterize the Jordan $\ast$-isomorphisms. These results unify and enrich previous works on algebra $\ast$-homomorphisms on $C(X,\mathcal{A})$ and $\mathrm{Lip}(X,\mathcal{A})$ for several concrete examples of $\mathcal{A}$.