Simple nuclear <i>C</i> *-algebras not isomorphic to their opposites
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Significance The Hilbert space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mi mathvariant="normal">ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> is the (usually infinite-dimensional) modification of our standard three-dimensional space. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -algebras are suitably closed algebras of linear operators on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mi mathvariant="normal">ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> . The algebras of complex <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mpadded width="+1.7pt"> <mml:mi>n</mml:mi> </mml:mpadded> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> matrices are the simplest examples of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -algebras. The opposite of a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -algebra is the algebra in which the direction of the multiplication is reversed. Although every matrix algebra is isomorphic to its opposite, we construct an inductive limit of matrix algebras not isomorphic to its opposite. This algebra is an example of a simple amenable <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -algebra not isomorphic to its opposite. Our examples can have exactly <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>n</mml:mi> </mml:math> inequivalent irreducible representations for any <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>n</mml:mi> </mml:math> , showing that Glimm’s dichotomy can fail for simple nonseparable <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mi>C</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -algebras.
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 | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.003 | 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