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

DOI:
https://doi.org/10.1090/S0002-9939-99-04819-4

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 -algebra onto another commutative -algebra with connected infinite Gelfand space is either linear or anti-linear.
- (ii)
- A ring automorphism of is either linear or anti-linear.
- (iii)
- , and the disc algebra are neither ring homomorphic images of nor .

**1.**B. E. Johnson,*The uniqueness of the (complete) norm topology*, Bull. Amer. Math. Soc.**73**(1967), 537-539. MR**35:2142****2.**M. Kuczma,*An introduction to the theory of functional equations and inequalities*, Panstwowe Wydawnictwo Naukowe, Warszawa, 1985. MR**86i:39008****3.**R. Larsen,*An introduction to the theory of multipliers*, Springer-Verlag, Berlin, 1971. MR**55:8695****4.**R. Larsen,*Banach algebras*, Marcel Dekker, Inc., New York, 1973. MR**58:7010****5.**L. Molnar,*The range of a ring homomorphism from a commutative -algebra*, Proc. Amer. Math. Soc.**124**(1996), 1789-1794. MR**96h:46090****6.**S.-E. Takahasi and O. Hatori,*Commutative Banach algebras and BSE-inequalities*, Math. Japonica**37**(1992), 607-614. MR**93h:46069**

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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:
https://doi.org/10.1090/S0002-9939-99-04819-4

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