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

 

Extensions of endomorphisms of $ C(X)$


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


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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: http://dx.doi.org/10.1090/S0002-9939-06-08441-3
PII: S 0002-9939(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.