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The range of a ring homomorphism
from a commutative $C^{*}$-algebra

Author: Lajos Molnár
Journal: Proc. Amer. Math. Soc. 124 (1996), 1789-1794
MSC (1991): Primary 46J05, 46E25
MathSciNet review: 1307551
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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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Additional Information

Lajos Molnár
Affiliation: Institute of Mathematics, Lajos Kossuth University, 4010 Debrecen, P.O.Box 12, Hungary

Keywords: Ring homomorphism, commutative Banach algebra, Gelfand representation
Received by editor(s): November 21, 1994
Additional Notes: Research partially supported by the Hungarian National Research Science Foundation, Operating Grant Number OTKA 1652 and K&H Bank Ltd., Universitas Foundation.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society