Residual finite-dimensionality and realizations of approximate rigidity phenomena for operator algebras
Bibliographic record
Abstract
Representations on a Hilbert space are a common tool for understanding both C*-algebras and (non self-adjoint) operator algebras. For operator algebras, a major theme has been to capitalize on the structural theory of an enveloping C*-algebra (or C*-cover) to better understand the underlying subalgebra. This philosophy has spanned several advancements, yet there is a subtlety involved in the process: the C*-algebra can vary greatly depending on the choice of representation of the operator algebra. Thus, it becomes advantageous to leverage information from several C*-covers to better understand the underlying operator algebra. Here, we focus on interactions between an operator algebra and its C*-covers, with an emphasis on sufficiency and maximality conditions for two fundamental classes of representations: the finite-dimensional representations and the non-commutative Choquet boundary, respectively. Both families offer separate advantages because the former class is more tractable, yet the latter induces a minimal representation due to work of Arveson. In Chapter 3, we analyze the residual finite-dimensionality of the maximal C*-cover of an operator algebra. To this end, we consider several C*-covers that are formed through families of finite-dimensional representations and compare these C*-covers with the maximal C*-cover. Along the way, we generalize Hadwin's characterization of separable residually finite-dimensional C*-algebras. In Chapter 4, we study maximality conditions on the non-commutative Choquet boundary. A conjecture of Arveson asserts that maximality conditions imply a rigidity property that is, a priori, much stronger. In this chapter, we uncover a significant localization procedure that generalizes several past attempts at Arveson's conjecture. This localization procedure is also applied to another conjecture of Arveson that concerns quotient modules of the Drury-Arveson space. Arveson's rigidity conjecture is originally inspired by a development in approximation theory due to Šaškin. In Chapter 5, we achieve one non-commutative analogue to Šaškin's theorem. In the setting of classical function theory, this encodes a maximality condition for the Choquet boundary with a rigidity property. We find that a similar phenomenon is still true for a large class of C*-algebras.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| 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 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".