Relative weak compactness of orbits in Banach spaces associated with locally compact groups

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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains an infinite <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>-set such that for every neighbourhood <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> of the identity of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove normal upper Gamma With caret"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\widehat \Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the interpolation (except at a finite subset depending 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>) can be done using at most 4 point masses. iv) A new proof that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B left-parenthesis upper G right-parenthesis subset-of upper W upper A upper P left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>⊂</mml:mo> <mml:mi>W</mml:mi> <mml:mi>A</mml:mi> <mml:mi>P</mml:mi> <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">B(G)\subset WAP(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for abelian groups is given that identifies the weak limits of translates of Fourier-Stieltjes transforms. v) Analogous results for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript o Baseline left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <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">C_o(G)</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 A Subscript p Baseline left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>p</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">A_p(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="upper M Subscript p Baseline left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>p</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">M_p(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are given. vi) Semigroup compactifications of groups are studied, both abelian and non-abelian: the weak* closure of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper G With caret"> <mml:seman