Regularity and Algebras of Analytic Functions in Infinite Dimensions

Abstract

A Banach space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is known to be Arens regular if every continuous linear mapping from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E prime"> <mml:semantics> <mml:msup> <mml:mi>E</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">E’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is weakly compact. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding="application/x-tex">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an open subset of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript b Baseline left-parenthesis upper U right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>U</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H_b(U)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the algebra of analytic functions on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding="application/x-tex">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which are bounded on bounded subsets of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding="application/x-tex">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula> lying at a positive distance from the boundary of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U period"> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">U.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> We endow <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript b Baseline left-parenthesis upper U right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>U</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H_b(U)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with the usual Fréchet topology. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript b Baseline left-parenthesis upper U right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>U</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">M_b(U)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes the set of continuous homomorphisms <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi colon upper H Subscript b Baseline left-parenthesis upper U right-parenthesis right-arrow double-struck upper C"> <mml:semantics> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>U</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">→</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\phi :H_b(U) \to \mathbb {C}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We study the relation between the Arens regularity of the space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the structure of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript b Baseline left-parenthesis upper U right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>U</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">M_b(U)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Locations

• Transactions of the American Mathematical Society - View - PDF