Norming algebras and automatic complete boundedness of isomorphisms of operator algebras
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Bibliographic record
Abstract
We combine the notion of norming algebra introduced by Pop, Sinclair and Smith with a result of Pisier to show that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A 1"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {A}_1</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="script upper A 2"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {A}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are operator algebras, then any bounded epimorphism of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A 1"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {A}_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> onto <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A 2"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {A}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is completely bounded provided that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A 2"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {A}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains a norming <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> -subalgebra. We use this result to give some insights into Kadison’s Similarity Problem: we show that every faithful bounded homomorphism of a <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 on a Hilbert space has completely bounded inverse, and show that a bounded representation of a <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 is similar to a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="asterisk"> <mml:semantics> <mml:mo> ∗ </mml:mo> <mml:annotation encoding="application/x-tex">*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -representation precisely when the image operator algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda"> <mml:semantics> <mml:mi> λ </mml:mi> <mml:annotation encoding="application/x-tex">\lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -norms itself. We give two applications to isometric isomorphisms of certain operator algebras. The first is an extension of a result of Davidson and Power on isometric isomorphisms of CSL algebras. Secondly, we show that an isometric isomorphism between subalgebras <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A Subscript i"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {A}_i</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"> <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> -diagonals <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis script upper C Subscript i Baseline comma script upper D Subscript i Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:m
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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 | 0.002 | 0.002 |
| 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 | 0.003 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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 it