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Here we introduce, for the data of a commutative ring $\\ell$, matrices $A,B$ as above and $C$ of the same size with coefficients in the group $\\mathcal{U}(\\ell)$ of invertible elements, an $\\ell$-algebra $\\mathcal{O}_{A,B}^C$, the twisted Kats"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2309.14325","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2023-09-25T17:52:31Z","cross_cats_sorted":["math.KT","math.QA","math.RA","math.RT"],"title_canon_sha256":"5386bd615583cc3e154f211d2a345e58030d6f73dd2ee746bf8d6d1f487c95fa","abstract_canon_sha256":"6e56796b5946f6d937c7b91de37c473ead781568ea13d76c9637493668fc4bcf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:00:45.457215Z","signature_b64":"16nhdF1N7R9UFYsRt17e8mwdtJrabUEmw3wR7dJ8l0OV5MQZx/HQYsUxTz0Ea0EfQJtLM7dHc31SS+n7WGX8CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c8a4c86c1f2c7a0e55a4de74fd20d07db454836c38a5ce1f2cfa4d2787929cf0","last_reissued_at":"2026-07-05T07:00:45.456802Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:00:45.456802Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exel-Pardo algebras with a twist","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.KT","math.QA","math.RA","math.RT"],"primary_cat":"math.OA","authors_text":"Guillermo Corti\\~nas","submitted_at":"2023-09-25T17:52:31Z","abstract_excerpt":"Katsura associated a $C^*$-algebra $C^*_{A,B}$ to integral matrices $A\\ge 0$ and $B$ of the same size, gave sufficient conditions on $(A,B)$ making it simple purely infinite (SPI), and proved that any separable $C^*$-algebra $KK$-isomorphic to a cone of an element $\\xi\\in KK(C(S^1)^n,C(S^1)^n)$ in Kasparov's $KK$ is $KK$-isomorphic to an SPI $C^*_{A,B}$. Here we introduce, for the data of a commutative ring $\\ell$, matrices $A,B$ as above and $C$ of the same size with coefficients in the group $\\mathcal{U}(\\ell)$ of invertible elements, an $\\ell$-algebra $\\mathcal{O}_{A,B}^C$, the twisted Kats"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.14325","kind":"arxiv","version":2},"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/2309.14325/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2309.14325","created_at":"2026-07-05T07:00:45.456869+00:00"},{"alias_kind":"arxiv_version","alias_value":"2309.14325v2","created_at":"2026-07-05T07:00:45.456869+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2309.14325","created_at":"2026-07-05T07:00:45.456869+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZCSMQ3A7FR5A","created_at":"2026-07-05T07:00:45.456869+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZCSMQ3A7FR5A4VNE","created_at":"2026-07-05T07:00:45.456869+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZCSMQ3A7","created_at":"2026-07-05T07:00:45.456869+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2412.15112","citing_title":"Homology of Steinberg algebras","ref_index":12,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZCSMQ3A7FR5A4VNE3Z2P2IGQPW","json":"https://pith.science/pith/ZCSMQ3A7FR5A4VNE3Z2P2IGQPW.json","graph_json":"https://pith.science/api/pith-number/ZCSMQ3A7FR5A4VNE3Z2P2IGQPW/graph.json","events_json":"https://pith.science/api/pith-number/ZCSMQ3A7FR5A4VNE3Z2P2IGQPW/events.json","paper":"https://pith.science/paper/ZCSMQ3A7"},"agent_actions":{"view_html":"https://pith.science/pith/ZCSMQ3A7FR5A4VNE3Z2P2IGQPW","download_json":"https://pith.science/pith/ZCSMQ3A7FR5A4VNE3Z2P2IGQPW.json","view_paper":"https://pith.science/paper/ZCSMQ3A7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2309.14325&json=true","fetch_graph":"https://pith.science/api/pith-number/ZCSMQ3A7FR5A4VNE3Z2P2IGQPW/graph.json","fetch_events":"https://pith.science/api/pith-number/ZCSMQ3A7FR5A4VNE3Z2P2IGQPW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZCSMQ3A7FR5A4VNE3Z2P2IGQPW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZCSMQ3A7FR5A4VNE3Z2P2IGQPW/action/storage_attestation","attest_author":"https://pith.science/pith/ZCSMQ3A7FR5A4VNE3Z2P2IGQPW/action/author_attestation","sign_citation":"https://pith.science/pith/ZCSMQ3A7FR5A4VNE3Z2P2IGQPW/action/citation_signature","submit_replication":"https://pith.science/pith/ZCSMQ3A7FR5A4VNE3Z2P2IGQPW/action/replication_record"}},"created_at":"2026-07-05T07:00:45.456869+00:00","updated_at":"2026-07-05T07:00:45.456869+00:00"}