The Non-selfadjoint Approach to the Hao–Ng Isomorphism
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Bibliographic record
Abstract
Abstract In an earlier work, the authors proposed a non-selfadjoint approach to the Hao–Ng isomorphism problem for the full crossed product, depending on the validity of two conjectures stated in the broader context of crossed products for operator algebras. By work of Harris and Kim, we now know that these conjectures in the generality stated may not always be valid. In this paper we show that in the context of hyperrigid tensor algebras of $\mathrm{C}^*$-correspondences, each one of these conjectures is equivalent to the Hao–Ng problem. This is accomplished by studying the representation theory of non-selfadjoint crossed products of C$^*$-correspondence dynamical systems; in particular we show that there is an appropriate dilation theory. A large class of tensor algebras of $\mathrm{C}^*$-correspondences, including all regular ones, are shown to be hyperrigid. Using Hamana’s injective envelope theory, we extend earlier results from the discrete group case to arbitrary locally compact groups; this includes a resolution of the Hao–Ng isomorphism for the reduced crossed product and all hyperrigid $\mathrm{C}^*$-correspondences. A culmination of these results is the resolution of the Hao–Ng isomorphism problem for the full crossed product and all row-finite graph correspondences; this extends a recent result of Bedos, Kaliszewski, Quigg, and Spielberg.
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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 | 0.011 | 0.012 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.000 |
| Open science | 0.009 | 0.009 |
| Research integrity | 0.000 | 0.004 |
| Insufficient payload (model declined to judge) | 0.000 | 0.002 |
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