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

TL;DR

This paper characterizes Jordan *-homomorphisms on spaces of continuous and Lipschitz maps into C*-algebras, showing they are represented as weighted composition operators and providing a characterization for primitive C*-algebras.

Contribution

It provides a comprehensive description of Jordan *-homomorphisms on function spaces into C*-algebras, extending previous results and unifying various cases.

Findings

Jordan *-homomorphisms are represented as weighted composition operators.

Characterization of Jordan *-isomorphisms for primitive C*-algebras.

Unification of previous results on algebra *-homomorphisms.

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 $\mathcal{A}$.