The đŸ-theory of Toeplitz đ¶*-algebras of right-angled Artin groups
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Bibliographic record
Abstract
Toeplitz <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mo> â </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">C^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebras of right-angled Artin groups were studied by Crisp and Laca. They are a special case of the Toeplitz <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mo> â </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">C^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebras <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper T left-parenthesis upper G comma upper P right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">T</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {T}(G, P)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> associated with quasi-lattice ordered groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper G comma upper P right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(G, P)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> introduced by Nica. Crisp and Laca proved that the so-called âboundary quotientsâ <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript upper Q Superscript asterisk Baseline left-parenthesis normal upper Gamma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>Q</mml:mi> <mml:mo> â </mml:mo> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Î </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C^*_Q(\Gamma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk Baseline left-parenthesis normal upper Gamma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mo> â </mml:mo> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Î </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> are simple and purely infinite. For a certain class of finite graphs <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"> Î </mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript upper Q Superscript asterisk Baseline left-parenthesis normal upper Gamma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>Q</mml:mi> <mml:mo> â </mml:mo> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Î </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C^*_Q(\Gamma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be represented as a full corner of a crossed product of an appropriate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mo> â </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">C^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -subalgebra of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript upper Q Superscript asterisk Baseline left-parenthesis normal upper Gamma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>Q</mml:mi> <mml:mo> â </mml:mo> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Î </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C^*_Q(\Gamma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> built by using <inline-formula content-type="math/mathml"> <mml
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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 | 0.001 |
| Science and technology studies | 0.000 | 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