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• W1493809586 abstract "The following results on uniqueness of invariant means are shown: (i) Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper G> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>G</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a connected almost simple algebraic group defined over <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper Q> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Q</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Assume that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper G left-parenthesis double-struck upper R right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>G</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {G}(mathbb {R})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the group of the real points in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper G> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>G</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, is not compact. Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=application/x-tex>p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a prime, and let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper G left-parenthesis double-struck upper Z Subscript p Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>G</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Z</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {G}({mathbb {Z}}_{p})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the compact <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=application/x-tex>p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic Lie group of the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper Z Subscript p> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Z</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>{mathbb {Z}}_{p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>–points in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper G> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>G</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Then the normalized Haar measure on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper G left-parenthesis double-struck upper Z Subscript p Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>G</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Z</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {G}({mathbb {Z}}_{p})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the unique invariant mean on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Superscript normal infinity Baseline left-parenthesis double-struck upper G left-parenthesis double-struck upper Z Subscript p Baseline right-parenthesis right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>G</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Z</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>L^{infty }(mathbb {G}({mathbb {Z}}_{p}))</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. (ii) Let <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> be a semisimple Lie group with finite centre and without compact factors, and let <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<!-- Γ --></mml:mi> <mml:annotation encoding=application/x-tex>Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a lattice in <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>. Then integration against the <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>–invariant probability measure on the homogeneous space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G slash normal upper Gamma> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>G/Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the unique <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<!-- Γ --></mml:mi> <mml:annotation encoding=application/x-tex>Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>–invariant mean on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Superscript normal infinity Baseline left-parenthesis upper G slash normal upper Gamma right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>G</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>L^{infty } (G/Gamma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W1493809586 created "2016-06-24" @default.
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• W1493809586 date "1998-01-01" @default.
• W1493809586 modified "2023-09-25" @default.
• W1493809586 title "On uniqueness of invariant means" @default.
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