Tensor products of commutative Banach algebras

U. B. Tewari, M. Dutta and Shobha Madan

International Journal of Mathematics and Mathematical Sciences, 1982, vol. 5, 1-10

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.

Date: 1982
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:867478

DOI: 10.1155/S0161171282000477

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