Ergodic equivalence relations, cohomology, and von Neumann algebras. I

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X comma script upper B right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(X,\mathcal {B})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a standard Borel space, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R subset-of upper X times upper X"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>X</mml:mi> <mml:mo>×</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">R \subset X \times X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> an equivalence relation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="element-of script upper B times script upper B"> <mml:semantics> <mml:mrow> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> <mml:mo>×</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\in \mathcal {B} \times \mathcal {B}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Assume each equivalence class is countable. Theorem 1: <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="there-exists"> <mml:semantics> <mml:mi mathvariant="normal">∃</mml:mi> <mml:annotation encoding="application/x-tex">\exists</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a countable group <italic>G</italic> of Borel isomorphisms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X comma script upper B right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(X,\mathcal {B})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> so that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R equals StartSet left-parenthesis x comma g x right-parenthesis colon g element-of upper G EndSet"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>g</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>G</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">R = \{ (x,gx):g \in G\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. <italic>G</italic> is far from unique. However, notions like invariance and quasi-invariance and <italic>R</italic>-<italic>N</italic> derivatives of measures depend only on <italic>R</italic>, not the choice of <italic>G</italic>. We develop some of the ideas of Dye [1], [2] and Krieger [1]-[5] in a fashion explicitly avoiding any choice of <italic>G</italic>; we also show the connection with virtual groups. A notion of “module over <italic>R</italic>” is defined, and we axiomatize and develop a cohomology theory for <italic>R</italic> with coefficients in such a module. Surprising application (contained in Theorem 7): let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha comma beta"> <mml:semantics> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha ,\beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be rationally independent irrationals on the circle <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper T"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {T}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <italic>f</italic> Borel: <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper T right-arrow double-struck upper T"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mo stretchy="false">→</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {T} \to \mathbb {T}</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="there-exists"> <mml:semantics> <mml:mi mathvariant="normal">∃</mml:mi> <mml:annotation encoding="application/x-tex">\exists</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Borel <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g comma h colon double-struck upper T right-arrow double-struck upper T"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>h</mml:mi> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mo stretchy="false">→</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">g,h:\mathbb {T} \to \mathbb {T}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x right-parenthesis equals left-parenthesis g left-parenthesis a x right-parenthesis slash g left-parenthesis x right-parenthesis right-parenthesis left-parenthesis h left-parenthesis beta x right-parenthesis slash h left-parenthesis x right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>h</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>β</mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>h</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f(x) = (g(ax)/g(x))(h(\beta x)/h(x))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a.e. The notion of “skew product action” is generalized to our context, and provides a setting for a generalization of the Krieger invariant for the <italic>R</italic>-<italic>N</italic> derivative of an ergodic transformation: we define, for a cocycle <italic>c</italic> on <italic>R</italic> with values in the group <italic>A</italic>, a subgroup of <italic>A</italic> depending only on the cohomology class of <italic>c</italic>, and in Theorem 8 identify this with another subgroup, the “normalized proper range” of <italic>c</italic>, defined in terms of the skew action. See also Schmidt [1].