Tensor Products of Commutative Banach Algebras

Let AI, be commutative semlslmple Banach algebras and 1 02 A2 be their projective tensor product. We prove that, if 10 2 is a group algebra (measure algebra) of a locally compact abelian group, then so are A 1 and A2. As a consequence, we prove that, if G is a locally compact abelian group and A is a comutatlve semi-simple Banach algebra, then the Banach algebra LI(G,A) of A-valued Bochner integrable functions on G is a group algebra if and only if A is a group algebra. Furthermore, if A has the Radon-Nikodym property, then, the Banach algebra M(G,A) of A-valued regular Borel measures of bounded variation on G is a measure algebra only if A is a easure algebra.