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• W15828319 startingPage "2871" @default.
• W15828319 abstract "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 discrete group. Following Linnell and Schick one can define a continuous ring <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c left-parenthesis normal upper Gamma right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>c(Gamma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> associated with <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>. They proved that if the Atiyah Conjecture holds for a torsion-free 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<!-- Γ --></mml:mi> <mml:annotation encoding=application/x-tex>Gamma</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=c left-parenthesis normal upper Gamma right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>c(Gamma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a skew field. Also, if <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> has torsion and the Strong Atiyah Conjecture holds for <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>, then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c left-parenthesis normal upper Gamma right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>c(Gamma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a matrix ring over a skew field. The simplest example when the Strong Atiyah Conjecture fails is the lamplighter group <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Gamma equals double-struck upper Z 2 wreath-product double-struck upper Z> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>Γ<!-- Γ --></mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Z</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>≀<!-- ≀ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Z</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>Gamma =mathbb {Z}_2wr mathbb {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is known that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper C left-parenthesis double-struck upper Z 2 wreath-product double-struck upper Z right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>C</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Z</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>≀<!-- ≀ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Z</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {C}(mathbb {Z}_2wr mathbb {Z})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> does not even have a classical ring of quotients. Our main result is that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=application/x-tex>H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is amenable, then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c left-parenthesis double-struck upper Z 2 wreath-product upper H right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>Z</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>≀<!-- ≀ --></mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>c(mathbb {Z}_2wr H)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is isomorphic to a continuous ring constructed by John von Neumann in the 1930s." @default.
• W15828319 created "2016-06-24" @default.
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• W15828319 date "2016-03-22" @default.
• W15828319 modified "2023-09-27" @default.
• W15828319 title "Lamplighter groups and von Neumann‘s continuous regular ring" @default.
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• W15828319 doi "https://doi.org/10.1090/proc/13066" @default.
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