Extensions of endomorphisms of

Authors:
J. F. Feinstein and T. J. Oliver

Journal:
Proc. Amer. Math. Soc. **135** (2007), 109-117

MSC (2000):
Primary 46J10, 47B48

Published electronically:
June 28, 2006

MathSciNet review:
2280180

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a compact space we consider extending endomorphisms of the algebra to be endomorphisms of Arens-Hoffman and Cole extensions of . Given a non-linear, monic polynomial , with semi-simple, we show that if an endomorphism of extends to the Arens-Hoffman extension with respect to , then it also extends to the simple Cole extension with respect to . We show that the converse to this is false. For a locally connected, metric we characterize the algebraically closed in terms of the extendability of endomorphisms to Arens-Hoffman and to simple Cole extensions.

**1.**Richard Arens and Kenneth Hoffmann,*Algebraic extension of normed algebras*, Proc. Amer. Math. Soc.**7**(1956), 203–210. MR**0077901**, 10.1090/S0002-9939-1956-0077901-3**2.**B. J. Cole,*One-point parts and the peak point conjecture*, Ph.D. Thesis, Yale University, 1968.**3.**R. S. Countryman Jr.,*On the characterization of compact Hausdorff 𝑋 for which 𝐶(𝑋) is algebraically closed*, Pacific J. Math.**20**(1967), 433–448. MR**0208410****4.**Thomas Dawson,*A survey of algebraic extensions of commutative, unital normed algebras*, Function spaces (Edwardsville, IL, 2002) Contemp. Math., vol. 328, Amer. Math. Soc., Providence, RI, 2003, pp. 157–170. MR**1990397**, 10.1090/conm/328/05777**5.**Joel F. Feinstein and Herbert Kamowitz,*Endomorphisms of Banach algebras of infinitely differentiable functions on compact plane sets*, J. Funct. Anal.**173**(2000), no. 1, 61–73. MR**1760278**, 10.1006/jfan.1999.3555**6.**P. Galindo, T. W. Gamelin, and M. Lindström,*Composition operators on uniform algebras and the pseudohyperbolic metric*, J. Korean Math. Soc.**41**(2004), no. 1, 1–20. Satellite Conference on Infinite Dimensional Function Theory. MR**2048697**, 10.4134/JKMS.2004.41.1.001**7.**Osamu Hatori and Takeshi Miura,*On a characterization of the maximal ideal spaces of commutative 𝐶*-algebras in which every element is the square of another*, Proc. Amer. Math. Soc.**128**(2000), no. 4, 1185–1189. MR**1690991**, 10.1090/S0002-9939-99-05454-4**8.**Takeshi Miura and Kazuki Niijima,*On a characterization of the maximal ideal spaces of algebraically closed commutative 𝐶*-algebras*, Proc. Amer. Math. Soc.**131**(2003), no. 9, 2869–2876 (electronic). MR**1974344**, 10.1090/S0002-9939-02-06835-1**9.**John G. Hocking and Gail S. Young,*Topology*, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR**0125557****10.**Herbert Kamowitz,*Compact endomorphisms of Banach algebras*, Pacific J. Math.**89**(1980), no. 2, 313–325. MR**599123****11.**Herbert Kamowitz and Stephen Scheinberg,*Some properties of endomorphisms of Lipschitz algebras*, Studia Math.**96**(1990), no. 3, 255–261. MR**1067312****12.**Udo Klein,*Kompakte multiplikative Operatoren auf uniformen Algebren*, Mitt. Math. Sem. Giessen**232**(1997), iv+120 (German). MR**1479364**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
46J10,
47B48

Retrieve articles in all journals with MSC (2000): 46J10, 47B48

Additional Information

**J. F. Feinstein**

Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, England

Email:
Joel.Feinstein@nottingham.ac.uk

**T. J. Oliver**

Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, England

Email:
Thomas.Oliver@maths.nottingham.ac.uk

DOI:
https://doi.org/10.1090/S0002-9939-06-08441-3

Received by editor(s):
October 14, 2004

Received by editor(s) in revised form:
July 21, 2005

Published electronically:
June 28, 2006

Additional Notes:
The second author would like to thank the EPSRC for providing support for this research

Communicated by:
N. Tomczak-Jaegermann

Article copyright:
© Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.