A generalized Powers averaging property for commutative crossed products
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Bibliographic record
Abstract
We prove a generalized version of Powers’ averaging property that characterizes simplicity of reduced crossed products <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis upper X right-parenthesis right-normal-factor-semidirect-product Subscript lamda Baseline upper G"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mo>⋊</mml:mo> <mml:mi>λ</mml:mi> </mml:msub> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">C(X) \rtimes _\lambda G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <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 countable discrete group, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a compact Hausdorff space which <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> acts on minimally by homeomorphisms. As a consequence, we generalize results of Hartman and Kalantar on unique stationarity to the state space of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis upper X right-parenthesis right-normal-factor-semidirect-product Subscript lamda Baseline upper G"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mo>⋊</mml:mo> <mml:mi>λ</mml:mi> </mml:msub> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">C(X) \rtimes _\lambda G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and to Kawabe’s generalized space of amenable subgroups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S u b Subscript a Baseline left-parenthesis upper X comma upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>Sub</mml:mi> <mml:mi>a</mml:mi> </mml:msub> <mml:mo></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {Sub}_a(X,G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This further lets us generalize a result of the first named author and Kalantar on simplicity of intermediate C*-algebras. We prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis upper Y right-parenthesis subset-of-or-equal-to upper C left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>⊆</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(Y) \subseteq C(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an inclusion of unital commutative <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>-C*-algebras with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> minimal and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis upper Y right-parenthesis right-normal-factor-semidirect-product Subscript lamda Baseline upper G"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mo>⋊</mml:mo> <mml:mi>λ</mml:mi> </mml:msub> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">C(Y) \rtimes _\lambda G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> simple, then any intermediate C*-algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfying <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis upper Y right-parenthesis right-normal-factor-semidirect-product Subscript lamda Baseline upper G subset-of-or-equal-to upper A subset-of-or-equal-to upper C left-parenthesis upper X right-parenthesis right-normal-factor-semidirect-product Subscript lamda Baseline upper G"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mo>⋊</mml:mo> <mml:mi>λ</mml:mi> </mml:msub> <mml:mi>G</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>A</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msub> <mml:mo>⋊</mml:mo> <mml:mi>λ</mml:mi> </mml:msub> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">C(Y) \rtimes _\lambda G \subseteq A \subseteq C(X) \rtimes _\lambda G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is simple.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it