Bounded extension property and 𝑝-sets

Abstract

The main result of this paper is a theorem which asserts that a closed subset of the compact Hausdorff space <italic>X</italic> is a <italic>p</italic>-set for a uniform algebra <italic>A</italic> on <italic>X</italic> if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S equals left-brace f element-of upper A semicolon upper R e f greater-than-or-slanted-equals 0 right-brace"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>A</mml:mi> <mml:mo>;</mml:mo> <mml:mi>Re</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>f</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>0</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">S = \{ f \in A;\operatorname {Re} f \geqslant 0\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has the so-called bounded extension property with respect to <italic>F</italic>. Similar results have been obtained by Bishop, Gamelin, Semadeni and the author.

Locations

• Proceedings of the American Mathematical Society - View - PDF