Completely isometric representations of 𝑀_{𝑐𝑏}𝐴(𝐺) and 𝑈𝐶𝐵(𝐺)*
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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 G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a locally compact group. It is shown that there exists a natural completely isometric representation of the completely bounded Fourier multiplier algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript c b Baseline upper A left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>c</mml:mi> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> <mml:mi>A</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">M_{cb}A(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which is dual to the representation of the measure algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">M(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper B left-parenthesis upper L 2 left-parenthesis upper G right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {B}(L_2(G))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The image algebras of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">M(G)</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 M Subscript c b Baseline upper A left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>c</mml:mi> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> <mml:mi>A</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">M_{cb}A(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C script upper B Superscript sigma Baseline left-parenthesis script upper B left-parenthesis upper L 2 left-parenthesis upper G right-parenthesis right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> σ </mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {CB}^{\sigma } (\mathcal {B}(L_2(G)))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are intrinsically characterized, and some commutant theorems are proved. It is also shown that for any amenable group <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> , there is a natural completely isometric representation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U upper C upper B left-parenthesis ModifyingAbove upper G With caret right-parenthesis Superscript asterisk"> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:mi>C</mml:mi> <mml:mi>B</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>G</mml:mi> <mml:mo stretchy="false"> ^ </mml:mo> </mml:mover> </mml:mrow> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ∗ </mml:mo> </mml:msup> </mml:mrow>
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| 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 it