On the character space of vector-valued Lipschitz algebras

We show that the character space of the‎ ‎vector-valued Lipschitz algebra $Lip^{\alpha}(X‎, ‎E)$ of order‎ ‎$\alpha$‎, ‎is homeomorphic to the cartesian product $X\times M_E$‎ ‎in the product topology‎, ‎where $X$ is a compact metric space and‎ ‎$E$ is a unital commutative Banach algebra‎. ‎We also characterize‎ ‎the form of each character on $Lip^{\alpha}(X‎, ‎E)$‎. ‎By appealing to the injective tensor product‎, ‎we then identify the‎ ‎character space of the vector-valued polynomial Lipschitz algebra‎ ‎$Lip_P^{\alpha}(X‎, ‎E)$‎, ‎generated by the polynomials on the‎ ‎compact space $X\subseteq \Bbb{C}^{n}$‎. ‎It is also shown that‎ ‎$Lip_P^{\alpha}(X‎, ‎E)$ is the injective tensor product‎ ‎$Lip_P^{\alpha}(X)\widehat{\otimes}_\epsilon E$‎. ‎Finally‎, ‎we characterize the form of each character on $Lip_{P}^{\alpha}(X‎, ‎E)$‎.