Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Studia Mathematica / All issues

Abstract

Let $\mathcal A$ be a unital $C^*$-algebra. We consider Jordan $*$-homomorphisms on $C(X, \mathcal A)$ and Jordan $*$-homomorphisms on 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 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 Lip$(X,\mathcal A)$ for several concrete examples of $\mathcal A$.