The range of a ring homomorphism from a commutative \(C^*\)-algebra. (English) Zbl 0857.46032
Summary: We prove that if a commutative semisimple Banach algebra \(\mathcal A\) is the range of a ring homomorphism from a commutative \(C^*\)-algebra, then \(\mathcal A\) is \(C^*\)-equivalent, i.e. there are a commutative \(C^*\)-algebra \(\mathcal B\) and a bicontinuous algebra isomorphism between \(\mathcal A\) and \(\mathcal B\). In particular, it is shown that the group algebras \(L^1(\mathbb{R})\), \(L^1(\mathbb{T})\) and the disc algebra \(A(\mathbb{D})\) are not ring homomorphic images of \(C^*\)-algebras.
MSC:
| 46J05 | General theory of commutative topological algebras |
| 46E25 | Rings and algebras of continuous, differentiable or analytic functions |
| 46L05 | General theory of \(C^*\)-algebras |
Keywords:
Gelfand representation; commutative semisimple Banach algebra; range of a ring homomorphism from a commutative \(C^*\)-algebra; \(C^*\)-equivalent; bicontinuous algebra isomorphism; ring homomorphic images of \(C^*\)-algebrasReferences:
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