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• W2109009695 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a Banach algebra, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a closed subspace of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A Superscript asterisk> <mml:semantics> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> <mml:annotation encoding=application/x-tex>A^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Y> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding=application/x-tex>Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a dual Banach space with predual <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Y Subscript asterisk> <mml:semantics> <mml:msub> <mml:mi>Y</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msub> <mml:annotation encoding=application/x-tex>Y_*</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi> <mml:semantics> <mml:mi>π<!-- π --></mml:mi> <mml:annotation encoding=application/x-tex>pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a continuous representation of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Y> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding=application/x-tex>Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We call <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi> <mml:semantics> <mml:mi>π<!-- π --></mml:mi> <mml:annotation encoding=application/x-tex>pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> subordinate to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if each coordinate function <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi Subscript y comma lamda Baseline element-of upper X> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>π<!-- π --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> </mml:mrow> </mml:msub> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>pi _{y,lambda }in X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, for all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=y element-of upper Y comma lamda element-of upper Y Subscript asterisk Baseline> <mml:semantics> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>Y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msub> <mml:mi>Y</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>yin Y, lambda in Y_*</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. If <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is topologically left (right) introverted and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Y> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding=application/x-tex>Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is reflexive, we show the existence of a natural bijection between continuous representations of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Y> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding=application/x-tex>Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> subordinate to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and normal representations of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X Superscript asterisk> <mml:semantics> <mml:msup> <mml:mi>X</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> <mml:annotation encoding=application/x-tex>X^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Y> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding=application/x-tex>Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a bounded approximate identity, then every weakly almost periodic functional on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a coordinate function of a continuous representation of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> subordinate to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper W upper A upper P left-parenthesis upper A right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mi>A</mml:mi> <mml:mi>P</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>WAP(A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that a function <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding=application/x-tex>f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a locally compact group <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is left uniformly continuous if and only if it is the coordinate function of the conjugate representation of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Superscript 1 Baseline left-parenthesis upper G right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>L^1(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, associated to some unitary representation of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We generalize the latter result to an arbitrary Banach algebra with bounded right approximate identity. We prove the functionals in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L upper U upper C left-parenthesis upper A right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mi>U</mml:mi> <mml:mi>C</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>LUC(A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are all coordinate functions of some norm continuous representation of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a dual Banach space." @default.
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• W2109009695 date "2015-04-24" @default.
• W2109009695 modified "2023-09-27" @default.
• W2109009695 title "Representations of Banach algebras subordinate to topologically introverted spaces" @default.
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