# Non-regularity for Banach function algebras

Studia Mathematica (2000)

• Volume: 141, Issue: 1, page 53-68
• ISSN: 0039-3223

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## Abstract

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Let A be a unital Banach function algebra with character space ${\Phi }_{A}$. For $x\in {\Phi }_{A}$, let ${M}_{x}$ and ${J}_{x}$ be the ideals of functions vanishing at x and in a neighbourhood of x, respectively. It is shown that the hull of ${J}_{x}$ is connected, and that if x does not belong to the Shilov boundary of A then the set $y\in {\Phi }_{A}:{M}_{x}\supseteq {J}_{y}$ has an infinite connected subset. Various related results are given.

## How to cite

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Feinstein, J., and Somerset, D.. "Non-regularity for Banach function algebras." Studia Mathematica 141.1 (2000): 53-68. <http://eudml.org/doc/216773>.

@article{Feinstein2000,
abstract = {Let A be a unital Banach function algebra with character space $Φ_\{A\}$. For $x ∈ Φ_\{A\}$, let $M_\{x\}$ and $J_\{x\}$ be the ideals of functions vanishing at x and in a neighbourhood of x, respectively. It is shown that the hull of $J_\{x\}$ is connected, and that if x does not belong to the Shilov boundary of A then the set $\{y ∈ Φ_\{A\}: M_\{x\} ⊇ J_\{y\}\}$ has an infinite connected subset. Various related results are given.},
author = {Feinstein, J., Somerset, D.},
journal = {Studia Mathematica},
keywords = {Hausdorff property; nonregular Banach function algebras; hulls; ideals},
language = {eng},
number = {1},
pages = {53-68},
title = {Non-regularity for Banach function algebras},
url = {http://eudml.org/doc/216773},
volume = {141},
year = {2000},
}

TY - JOUR
AU - Feinstein, J.
AU - Somerset, D.
TI - Non-regularity for Banach function algebras
JO - Studia Mathematica
PY - 2000
VL - 141
IS - 1
SP - 53
EP - 68
AB - Let A be a unital Banach function algebra with character space $Φ_{A}$. For $x ∈ Φ_{A}$, let $M_{x}$ and $J_{x}$ be the ideals of functions vanishing at x and in a neighbourhood of x, respectively. It is shown that the hull of $J_{x}$ is connected, and that if x does not belong to the Shilov boundary of A then the set ${y ∈ Φ_{A}: M_{x} ⊇ J_{y}}$ has an infinite connected subset. Various related results are given.
LA - eng
KW - Hausdorff property; nonregular Banach function algebras; hulls; ideals
UR - http://eudml.org/doc/216773
ER -

## References

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13. [13] D. W. B. Somerset, Minimal primal ideals in Banach algebras, Math. Proc. Cambridge Philos. Soc. 115 (1994), 39-52.
14. [14] D. W. B. Somerset, Ideal spaces of Banach algebras, Proc. London Math. Soc. (3) 78 (1999), 369-400. Zbl1027.46058
15. [15] G. Stolzenberg, The maximal ideal space of functions locally in an algebra, Proc. Amer. Math. Soc. 14 (1963), 342-345. Zbl0142.10102
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