International Journal of Mathematics and Mathematical Sciences
Volume 5 (1982), Issue 3, Pages 503-512
Tensor products of commutative Banach algebras
Department of Mathematics, Indian Institute of Technology, Kanpur 208016, U.P., India
Copyright © 1982 U. B. Tewari et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let , be commutative semisimple Banach algebras and be their projective tensor product. We prove that, if is a group algebra (measure algebra) of a locally compact abelian group, then so are and . As a consequence, we prove that, if is a locally compact abelian group and is a comutative semi-simple Banach algebra, then the Banach algebra of -valued Bochner integrable functions on is a group algebra if and only if is a group algebra. Furthermore, if has the Radon-Nikodym property, then the Banach algebra of -valued regular Borel measures of bounded variation on is a measure algebra only if is a measure algebra.