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

Let A be a unital C∗-algebra. We consider Jordan ∗-homomorphisms on C(X,A) and Jordan ∗-homomorphisms on Lip(X,A). More precisely, for any unital C∗-algebra A, we prove that every Jordan ∗-homomorphism on C(X,A) and every Jordan ∗-homomorphism on Lip(X,A) is represented as a weighted composition operator by using the irreducible representations of A. In addition, when A1 and A2 are primitive C∗-algebras, we characterize the Jordan ∗-isomorphisms. These results unify and enrich previous works on algebra ∗-homomorphisms on C(X,A) and Lip(X,A) for several concrete examples of A.