Preservation of the joint essential matricial range
Bibliographic record
Abstract
Let A = ( A 1 , ⋯ , A m ) be an m-tuple of elements of a unital C ∗ -algebra A and let M q denote the set of q × q complex matrices. The joint q-matricial range W q ( A ) is the set of ( B 1 , ⋯ , B m ) ∈ M q m such that B j = Φ ( A j ) for some unital completely positive linear map Φ : A → M q . When A = B ( H ) , where B ( H ) is the algebra of bounded linear operators on the Hilbert space H, the joint spatial q-matricial range W s q ( A ) of A is the set of ( B 1 , ⋯ , B m ) ∈ M q m for which there is a q-dimensional subspace V of H such that B j is the compression of A j to V for j = 1 , ⋯ , m . Suppose that K ( H ) is the set of compact operators in B ( H ) . The joint essential spatial q-matricial range is defined as W e s s q ( A ) = ∩ { cl ( W s q ( A 1 + K 1 , ⋯ , A m + K m ) ) : K 1 , ⋯ , K m ∈ K ( H ) } , where cl ( T ) denotes the closure of the set T. Let π be the canonical surjection from B ( H ) to the Calkin algebra B ( H ) / K ( H ) . We prove that W e s s q ( A ) = W q ( π ( A ) ) , where π ( A ) = ( π ( A 1 ) , ⋯ , π ( A m ) ) . Furthermore, for any positive integer N, we prove that there are self-adjoint compact operators K 1 , ⋯ , K m such that cl W s q ( A 1 + K 1 , ⋯ , A m + K m ) = W e s s q ( A ) for all q ∈ { 1 , ⋯ , N } . These results generalize those of Narcowich–Ward and Smith–Ward, obtained in the m = 1 case, and also generalize a result of Müller obtained in case m ⩾ 1 and q = 1 . Furthermore, if W e s s 1 ( A ) is a simplex in R m , then we prove that there are self-adjoint compact operators K 1 , ⋯ , K m such that cl ( W s q ( A 1 + K 1 , ⋯ , A m + K m ) ) = W e s s q ( A ) for all positive integers q.
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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) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.002 | 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".