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 $xtimes m_e$ 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 $xsubseteq bbb{c}^{n}$. 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 $lip_{p}^{alpha}(x, e)$.