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

 

Sufficient conditions for a linear functional to be multiplicative


Authors: K. Seddighi and M. H. Shirdarreh Haghighi
Journal: Proc. Amer. Math. Soc. 129 (2001), 2385-2393
MSC (2000): Primary 46J20; Secondary 46J10
Published electronically: January 17, 2001
MathSciNet review: 1823923
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Abstract:

A commutative Banach algebra $\mathcal{A}$ is said to have the $P(k,n)$property if the following holds: Let ${{M}}$ be a closed subspace of finite codimension $n$ such that, for every $x\in {{M}}$, the Gelfand transform $\hat{x}$ has at least $k$ distinct zeros in $\Delta(\mathcal{A})$, the maximal ideal space of $\mathcal{A}$. Then there exists a subset $Z$ of $\Delta(\mathcal{A})$of cardinality $k$ such that $\hat{{M}}$ vanishes on $Z$, the set of common zeros of ${{M}}$. In this paper we show that if $X\subset \mathbf{C}$ is compact and nowhere dense, then $R(X)$, the uniform closure of the space of rational functions with poles off $X$, has the $P(k,n)$ property for all $k,n\in \mathbf{N}$. We also investigate the $P(k,n)$ property for the algebra of real continuous functions on a compact Hausdorff space.


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Additional Information

K. Seddighi
Affiliation: Department of Mathematics, Shiraz University, Shiraz 71454, Iran

M. H. Shirdarreh Haghighi
Affiliation: Department of Mathematics, Shiraz University, Shiraz 71454, Iran
Email: shir@sun01.susc.ac.ir

DOI: http://dx.doi.org/10.1090/S0002-9939-01-05720-3
PII: S 0002-9939(01)05720-3
Keywords: Multiplicative linear functional, the $P(k,n)$ property, Banach algebra, maximal ideal
Received by editor(s): January 31, 1999
Received by editor(s) in revised form: December 17, 1999
Published electronically: January 17, 2001
Additional Notes: This research was partially supported by a grant from IPM, The Institute for Studies in Theoretical Physics and Mathematics.
Communicated by: David R. Larson
Article copyright: © Copyright 2001 American Mathematical Society