Approximate unitary equivalence in simple 𝐶*-algebras of tracial rank one
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Bibliographic record
Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a unital AH-algebra and let <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> be a unital separable simple <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> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">C^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebra with tracial rank no more than one. Suppose that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi comma psi colon upper C right-arrow upper A"> <mml:semantics> <mml:mrow> <mml:mi> ϕ </mml:mi> <mml:mo>,</mml:mo> <mml:mi> ψ </mml:mi> <mml:mo>:</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\phi , \psi : C\to A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are two unital monomorphisms. With some restriction on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C comma"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C,</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="phi"> <mml:semantics> <mml:mi> ϕ </mml:mi> <mml:annotation encoding="application/x-tex">\phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="psi"> <mml:semantics> <mml:mi> ψ </mml:mi> <mml:annotation encoding="application/x-tex">\psi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are approximately unitarily equivalent if and only if <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column left-bracket phi right-bracket 2nd Column a m p semicolon equals 3rd Column a m p semicolon left-bracket psi right-bracket in upper K upper L left-parenthesis upper C comma upper A right-parenthesis comma 2nd Row 1st Column tau ring phi 2nd Column a m p semicolon equals 3rd Column a m p semicolon tau ring psi for all tracial states of upper A and 3rd Row 1st Column phi Superscript double-dagger 2nd Column a m p semicolon equals 3rd Column a m p semicolon psi Superscript double-dagger Baseline comma EndLayout"> <mml:semantics> <mml:mtable columnalign="right center left" rowspacing="3pt" columnspacing="0 thickmathspace" side="left" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:mo stretchy="false">[</mml:mo> <mml:mi> ϕ </mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mo>=</mml:mo> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mi> ψ </mml:mi> <mml:mo stretchy="false">]</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mspace width="thinmathspace"/> <mml:mspace width="thinmathspace"/> <mml:mtext>in</mml:mtext> <mml:mspace width="thinmathspace"/> <mml:mspace width="thinmathspace"/> <mml:mspace width="thinmathspace"/> <mml:mi>K</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>,</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi> τ </mml:mi> <mml:mo> ∘ </mml:mo> <mml:mi> ϕ </mml:mi> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mo>=</mml:mo> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mi> τ </mml:mi> <mml:mo> ∘ </mml:mo> <mml:mi> ψ </mml:mi> <mml:mtext> </mml:mtext> <mml:mtext>for all tracial states of</mml:mtext> <mml:mspace width="thinmathspace"/> <mml:mspace width="thinmathspace"/> <mml:mspace width="thinmathspace"/> <mml:mi>A</mml:mi>
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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