The greatest regular subalgebra of certain Banach algebras of vector-valued functions

Given an arbitrary commutative complex Banach algebraA, it is shown that, for various classical Banach algebras ofA-valued functions, the greatest regular subalgebra consists precisely of those functions which map into the greatest regular subalgebra ofA. The main result covers the case of continuous and differentiable functions, Lipschitz functions, and Bochner integrable functions on a locally compact abelian group. The principal tools are from the theory of tensor products of Banach algebras.