New tensor products of C*-algebras and characterization of type I C*-algebras as rigidly symmetric C*-algebras
Bibliographic record
Abstract
Inspired by recent developments in the theory of Banach and operator algebras of locally compact groups, we construct several new classes of bifunctors <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper A comma upper B right-parenthesis right-arrow from bar upper A circled-times Subscript alpha Baseline upper B"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false"> ↦ </mml:mo> <mml:mi>A</mml:mi> <mml:msub> <mml:mo> ⊗ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> α </mml:mi> </mml:mrow> </mml:msub> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">(A,B)\mapsto A\otimes _{\alpha } B</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 A circled-times Subscript alpha Baseline upper B"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:msub> <mml:mo> ⊗ </mml:mo> <mml:mi> α </mml:mi> </mml:msub> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A\otimes _\alpha B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a cross norm completion of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A circled-dot upper B"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo> ⊙ </mml:mo> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A\odot B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for each pair of C*-algebras <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> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding="application/x-tex">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . For the first class of bifunctors considered <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper A comma upper B right-parenthesis right-arrow from bar upper A circled-times Subscript p Baseline upper B"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false"> ↦ </mml:mo> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mo> ⊗ </mml:mo> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">(A,B)\mapsto A{\otimes _p} B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ( <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 less-than-or-equal-to p less-than-or-equal-to normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo> ≤ </mml:mo> <mml:mi>p</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1\leq p\leq \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ), <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A circled-times Subscript p Baseline upper B"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mo> ⊗ </mml:mo> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A{\otimes _p} B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a Banach algebra cross-norm completion of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A circled-dot upper B"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo> ⊙ </mml:mo> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A\odot B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> constructed in a fashion similar to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -pseudofunctions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="PF Subscript p Superscript asterisk Baseline left-parenthesis upper G right-parenthesis"> <mml:semantics>
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How this classification was reachedexpand
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 | 0.000 |
| Bibliometrics | 0.000 | 0.003 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".