Relative weak compactness of orbits in Banach spaces associated with locally compact groups
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Abstract
We study analogues of weak almost periodicity in Banach spaces on locally compact groups. i) If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi> μ </mml:mi> <mml:annotation encoding="application/x-tex">\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a continous measure on the locally compact abelian 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> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper L Superscript normal infinity Baseline left-parenthesis mu right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi> μ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f\in L^\infty (\mu )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet gamma f colon gamma element-of ModifyingAbove upper G With caret EndSet"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi> γ </mml:mi> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi> γ </mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>G</mml:mi> <mml:mo> ^ </mml:mo> </mml:mover> </mml:mrow> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{\gamma f:\gamma \in \widehat G\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not relatively weakly compact. ii) If <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 a discrete abelian group and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of script l Superscript normal infinity Baseline left-parenthesis upper G right-parenthesis minus upper C Subscript o Baseline left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mi> ℓ </mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi class="MJX-variant" mathvariant="normal"> ∖ </mml:mi> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>o</mml:mi> </mml:msub> <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">f\in \ell ^\infty (G)\backslash C_o(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet gamma f colon gamma element-of upper E EndSet"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi> γ </mml:mi> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi> γ </mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>E</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{\gamma f:\gamma \in E\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not relatively weakly compact if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E subset-of ModifyingAbove upper G With caret"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo> ⊂ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>G</mml:mi> <mml:mo> ^ </mml:mo> </mml:mover> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">E\subset \widehat G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has non-empty interior. That result will follow from an existence theorem for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript o"> <mml:semantics> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>o</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">I_o</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -sets, as follows. iii) Every infinite subset of a discrete abelian group
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|---|---|---|
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