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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A structure of ring homomorphisms
on commutative Banach algebras


Authors: Sin-Ei Takahasi and Osamu Hatori
Journal: Proc. Amer. Math. Soc. 127 (1999), 2283-2288
MSC (1991): Primary 46J05, 46E25
Published electronically: April 9, 1999
MathSciNet review: 1486754
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Abstract | References | Similar Articles | Additional Information

Abstract: We give a structure theorem for a ring homomorphism of a commutative regular Banach algebra into another commutative Banach algebra. In particular, it is shown that:

(i)
A ring homomorphism of a commutative $\mathrm C^*$-algebra onto another commutative $\mathrm C^*$-algebra with connected infinite Gelfand space is either linear or anti-linear.
(ii)
A ring automorphism of $L^1(\boldsymbol{R}^N)$ is either linear or anti-linear.
(iii)
$C^n([a,b])$, $L^1(\boldsymbol{R}^N)$ and the disc algebra $A(D)$ are neither ring homomorphic images of $\ell^1(S)$ nor $L^p(G)$ $(1\le p<\infty,\ G\ \text{a compact abelian group})$.


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Additional Information

Sin-Ei Takahasi
Affiliation: Department of Basic Technology, Applied Mathematics and Physics, Yamagata University, Yonezawa 992-8510, Japan

Osamu Hatori
Affiliation: Department of Mathematical Science, Graduate School of Science and Technology, Niigata University, Niigata 950-2102, Japan
Email: hatori@math.sc.niigata-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04819-4
PII: S 0002-9939(99)04819-4
Keywords: Commutative Banach algebra, ring homomorphism, Gelfand transform, Fourier transform
Received by editor(s): May 29, 1997
Received by editor(s) in revised form: October 27, 1997
Published electronically: April 9, 1999
Additional Notes: The authors are partly supported by the Grants-in-Aid for Scientific Research, The Ministry of Education, Science and Culture, Japan
Dedicated: Dedicated to Professor Jyunji Inoue on his sixtieth birthday
Communicated by: David R. Larson
Article copyright: © Copyright 1999 American Mathematical Society