International Journal of Mathematics and Mathematical Sciences
Volume 5 (1982), Issue 3, Pages 503-512

Tensor products of commutative Banach algebras

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

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 A1, A2 be commutative semisimple Banach algebras and A1A2 be their projective tensor product. We prove that, if A1A2 is a group algebra (measure algebra) of a locally compact abelian group, then so are A1 and A2. 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 L1(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.