Tensor products of commutative Banach algebras
Let A1, A2 be commutative semisimple Banach algebras and A1⊗∂A2 be their projective tensor product. We prove that, if A1⊗∂A2 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...
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| Language: | English |
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Wiley
1982-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171282000477 |
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| author | U. B. Tewari M. Dutta Shobha Madan |
| author_facet | U. B. Tewari M. Dutta Shobha Madan |
| author_sort | U. B. Tewari |
| collection | DOAJ |
| description | Let A1, A2 be commutative semisimple Banach algebras and A1⊗∂A2 be their projective tensor product. We prove that, if A1⊗∂A2 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. |
| format | Article |
| id | doaj-art-69a62c2a457544a2b93db5298981d63b |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1982-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-69a62c2a457544a2b93db5298981d63b2025-02-03T01:29:11ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251982-01-015350351210.1155/S0161171282000477Tensor products of commutative Banach algebrasU. B. Tewari0M. Dutta1Shobha Madan2Department of Mathematics, Indian Institute of Technology, Kanpur 208016, U.P., IndiaDepartment of Mathematics, Indian Institute of Technology, Kanpur 208016, U.P., IndiaDepartment of Mathematics, Indian Institute of Technology, Kanpur 208016, U.P., IndiaLet A1, A2 be commutative semisimple Banach algebras and A1⊗∂A2 be their projective tensor product. We prove that, if A1⊗∂A2 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.http://dx.doi.org/10.1155/S0161171282000477commutative semisimple Banach algebraprojective tensor productmeasure algebralocally compact abelian groupRadon-Nikodym property. |
| spellingShingle | U. B. Tewari M. Dutta Shobha Madan Tensor products of commutative Banach algebras International Journal of Mathematics and Mathematical Sciences commutative semisimple Banach algebra projective tensor product measure algebra locally compact abelian group Radon-Nikodym property. |
| title | Tensor products of commutative Banach algebras |
| title_full | Tensor products of commutative Banach algebras |
| title_fullStr | Tensor products of commutative Banach algebras |
| title_full_unstemmed | Tensor products of commutative Banach algebras |
| title_short | Tensor products of commutative Banach algebras |
| title_sort | tensor products of commutative banach algebras |
| topic | commutative semisimple Banach algebra projective tensor product measure algebra locally compact abelian group Radon-Nikodym property. |
| url | http://dx.doi.org/10.1155/S0161171282000477 |
| work_keys_str_mv | AT ubtewari tensorproductsofcommutativebanachalgebras AT mdutta tensorproductsofcommutativebanachalgebras AT shobhamadan tensorproductsofcommutativebanachalgebras |