On Vector-Valued Banach Function Algebras

We consider vector-valued Banach function algebras on a compact Hausdorff space. Then, we define the subalgebras generated by vector-valued polynomials and rational functions, and determine their maximal ideal spaces and Šilov boundaries.We finally make use the results for a certain category of these algebras such as vector-valued Lipschitz algebras, vector-valued Dales–Davie algebras (algebras of vector-valued differentiable functions) and the algebras of vector-valued differentiable Lipschitz functions.