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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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
1982-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171282000477 |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |