{"paper":{"title":"The many faces of higher Hilbert spaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.CT","math.MP","math.OA"],"primary_cat":"math.QA","authors_text":"David Penneys, Giovanni Ferrer, Lukas M\\\"uller, Luuk Stehouwer","submitted_at":"2026-06-09T18:12:51Z","abstract_excerpt":"Finite-dimensional operator algebras can be viewed as $\\mathrm{C}^*$, $\\mathrm{W}^*$, or $\\mathrm{H}^*$-algebras, leading to different notions for their categories of modules and correspondence 2-categories. In this article, we show how these differences can be understood systematically using the notion of $G$-dagger category from arXiv:2403.01651 for different subgroups $G\\leq O(2)$. To do so, we first introduce $G$-Hermitian $2$-vector spaces using fixed points of a certain $O(2)$-action on $2\\mathsf{Vect}$. We then propose criteria for when such pairings are `positive', generalizing the pas"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11334","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.11334/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}