Jordan $*$-homomorphisms on the spaces of continuous maps taking values in $C^*$-algebras
Volume 269 / 2023
Studia Mathematica 269 (2023), 107-119
MSC: Primary 47B48; Secondary 46E40, 47B49.
DOI: 10.4064/sm220210-19-6
Published online: 28 September 2022
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 $\mathcal A$.
