Tensor Products and BSE-Algebras

In this paper, we investigate the BSE property of tensor product Ab B of commutative Banach algebras A and B. We show that if Ab B is a BSE-algebra, then A and B are BSE-algebras. In the special case, we investigate Banach algebras of vector-valued continuous functions on a com- pact Hausdor space X, and also vector-valued polynomial Lipschitz algebras on a compact plane set X.