AN ADDENDUM TO: UNITARILY INVARIANT LINEAR SPACES IN C∗-ALGEBRAS
Bibliographic record
Abstract
the set of all bounded linear operators acting on H. If H is finite di-mensional, then it is well known that the trace zero elements of B(H) coincides with set of commutators {[A,B] = AB−BA: A,B ∈ B(H)}. Furthermore, if A ∈ B(H) has trace zero, then we can find an orthonor-mal basis with respect to which the diagonal of the matrix for A consists only of zeros. The upper and lower triangular parts of this matrix give a decomposition of A as the sum of two (quasi)nilpotents in B(H). In infinite dimensions, the result is far less trivial. A result of Brown and Pearcy [1] states that an operator T ∈ B(H) is a commutator if and only if T 6 ∈ {λI +K: 0 6 = λ ∈ C, K compact in B(H)}. Fong and Sourour [4] have shown that this coincides with the set of operators which can be written as the sum of two quasinilpotents. As pointed out in a recent conference at Dalhousie University in Hal-ifax by P. Rosenthal, the connection between the sums of quasinilpo-tents and the set of commutators is less than obvious. The following continues to explore this connection in the setting of C∗-algebras. Notation. Let A be a C∗-algebra. We denote by • LP (A) the linear span of the projections in A; • [A,A] the linear span of the commutators in A; • LQ(A) the linear span of the quasinilpotents in A; • LN2(A) the linear span of the nilpotents of order 2 in A. When A possesses tracial states, we set sl(A) to be the intersection of the kernels of the tracial states on A. Observe that for each of the last three spaces above, the expression ‘linear span ’ can be replaced by ‘set of finite sums’.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.002 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.007 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.005 | 0.005 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.058 | 0.026 |
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 source (direct Gemma or distilled Codex), 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".