Isometries on Banach algebras of vector-valued maps

We propose a unified approach to the study of isometries on algebras of vector-valued Lipschitz maps and those of continuously differentiable maps by means of the notion of admissible quadruples. We describe isometries on function spaces of some admissible quadruples that take values in unital commutative $C^*$-algebras. As a consequence we confirm the statement of \cite[Example 8]{jp} on Lipschitz algebras and show that isometries on such algebras indeed take the canonical form.