Tensor products of commutative Banach algebras

Abstract

Let A 1 , A 2 be commutative semisimple Banach algebras and A 1 ⊗ ∂ A 2 be their projective tensor product. We prove that, if A 1 ⊗ ∂ A 2 is a group algebra (measure algebra) of a locally compact abelian group, then so are A 1 and A 2 . As a consequence, we prove that, if G is a locally compact abelian group and A is a comutative semi‐simple Banach algebra, then the Banach algebra L 1 ( 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 measure algebra.