Algebraic reflexivity of isometry groups of algebras of Lipschitz maps

We study groups of surjective linear isometries on Banach algebras of Lipschitz maps with values in some unital C⁎-algebras. In this paper, these spaces are endowed with the sum norm. For the case where the C⁎-algebras are commutative whose groups of all surjective isometries are algebraically reflexive, we prove that the group of all surjective isometries on the corresponding Banach algebra of Lipschitz maps are algebraically reflexive. We also prove that the group of unital surjective isometries between matrix-valued Lipschitz algebras are reflexive.