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

Abstract

Let $\mathcal{A}$ be a unital $C^{*}$-algebra. We consider Jordan $*$-homomorphisms on $C(X, \mathcal{A})$ and Jordan $*$-homomorphisms on $\operatorname{Lip}(X,\mathcal{A})$. More precisely, for any unital $C^{*}$-algebra $\mathcal{A}$, we prove that every Jordan $*$-homomorphism on $C(X, \mathcal{A})$ and every Jordan $*$-homomorphism on $\operatorname{Lip}(X,\mathcal{A})$ 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 $C^{*}$-algebras, we characterize the Jordan $*$-isomorphisms. These results unify and enrich previous works on algebra $*$-homomorphisms on $C(X, \mathcal{A})$ and $\operatorname{Lip}(X,\mathcal{A})$ for several concrete examples of $\mathcal{A}$.