The range of a ring homomorphism from a commutative 𝐶*-algebra

Molnár, Lajos

doi:10.1090/S0002-9939-96-03236-4

The range of a ring homomorphism from a commutative $C*$-algebra
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Proc. Amer. Math. Soc. 124 (1996), 1789-1794 Request permission

Abstract:

We prove that if a commutative semi-simple 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.

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