Sufficient conditions for a linear functional to be multiplicative
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- by K. Seddighi and M. H. Shirdarreh Haghighi PDF
- Proc. Amer. Math. Soc. 129 (2001), 2385-2393 Request permission
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.References
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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
- 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
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 2385-2393
- MSC (2000): Primary 46J20; Secondary 46J10
- DOI: https://doi.org/10.1090/S0002-9939-01-05720-3
- MathSciNet review: 1823923