On the character space of vector-valued Lipschitz algebras (Q274627)

Let \(X\) be a compact metric space and let \(E\) be a unital commutative Banach algebra. For \(0< \alpha \leq1\), consider \[ \mathrm{Lip}^\alpha(X,E) = \{f: X \rightarrow E: p(f) = \sup_{x \neq y~,x,y \in X} \frac{\| f(x)-f(y)\|}{d(x,y)^\alpha} < \infty\}, \] equipped with the norm \(\| f\| = \| f\|_{\infty} + p(f)\). This is a commutative Banach algebra and in this paper, the authors show that its character space is homeomorphic to the product space \(X \times M_E\) where \(M_E\) is the character space of \(E\).