Geodesic
Non-regularity for Banach function algebras
Studia Mathematica, Tome 141 (2000) no. 1, pp. 53-68
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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. 
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J. Feinstein; D. Somerset. Non-regularity for Banach function algebras. Studia Mathematica, Tome 141 (2000) no. 1, pp. 53-68. doi: 10.4064/sm-141-1-53-68

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