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We prove that if $A$ has property (TM), then the crossed product $A\\rtimes_\\alpha G$ has property (TM). As a corollary, if $A$ is an infinite-dimensional separable simple unital C*-algebra which has stable rank one and strict comparison, $\\alpha\\colon G\\rightarrow \\mathrm{Aut}(A)$ is an action of a finite group $G$ on $A$ with the weak tracial Rokhlin property, then $A\\rt"},"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":"2407.09867","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2024-07-13T12:35:44Z","cross_cats_sorted":[],"title_canon_sha256":"23cc904293e5185443cf539cb5559e8e6250f9646d10a2c9f46e9a16ade916cb","abstract_canon_sha256":"1fee56596abb2ce12121fa19abe976c2ba6c841f775467225f01655f72b54e1a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:43:35.886791Z","signature_b64":"vxt6jFT5bLNRO+YctCKqZtNFKaw9cTQFwnbSmU6ixamEjZeIQu+kkizXeN4rg1lRuWpu7YNWTYOmD6wFISNPCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"931bac81b5bd0df86c5fa21abef6b7167bb3921d70041e7e81198dd853a21545","last_reissued_at":"2026-07-05T08:43:35.886380Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:43:35.886380Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Stable rank for crossed products by finite group actions with the weak tracial Rokhlin property","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Xiaochun Fang, Zhongli Wang","submitted_at":"2024-07-13T12:35:44Z","abstract_excerpt":"Let $A$ be an infinite-dimensional stably finite simple unital C*-algebra, let $G$ be a finite group, and let $\\alpha\\colon G\\rightarrow \\mathrm{Aut}(A)$ be an action of $G$ on $A$ which has the weak tracial Rokhlin property. We prove that if $A$ has property (TM), then the crossed product $A\\rtimes_\\alpha G$ has property (TM). As a corollary, if $A$ is an infinite-dimensional separable simple unital C*-algebra which has stable rank one and strict comparison, $\\alpha\\colon G\\rightarrow \\mathrm{Aut}(A)$ is an action of a finite group $G$ on $A$ with the weak tracial Rokhlin property, then $A\\rt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.09867","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/2407.09867/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":"2407.09867","created_at":"2026-07-05T08:43:35.886437+00:00"},{"alias_kind":"arxiv_version","alias_value":"2407.09867v1","created_at":"2026-07-05T08:43:35.886437+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.09867","created_at":"2026-07-05T08:43:35.886437+00:00"},{"alias_kind":"pith_short_12","alias_value":"SMN2ZANVXUG7","created_at":"2026-07-05T08:43:35.886437+00:00"},{"alias_kind":"pith_short_16","alias_value":"SMN2ZANVXUG7Q3C7","created_at":"2026-07-05T08:43:35.886437+00:00"},{"alias_kind":"pith_short_8","alias_value":"SMN2ZANV","created_at":"2026-07-05T08:43:35.886437+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2412.10486","citing_title":"Uniform property $\\Gamma$ for Crossed products by group actions with the Rokhlin-type properties","ref_index":11,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SMN2ZANVXUG7Q3C7UINL55VXCZ","json":"https://pith.science/pith/SMN2ZANVXUG7Q3C7UINL55VXCZ.json","graph_json":"https://pith.science/api/pith-number/SMN2ZANVXUG7Q3C7UINL55VXCZ/graph.json","events_json":"https://pith.science/api/pith-number/SMN2ZANVXUG7Q3C7UINL55VXCZ/events.json","paper":"https://pith.science/paper/SMN2ZANV"},"agent_actions":{"view_html":"https://pith.science/pith/SMN2ZANVXUG7Q3C7UINL55VXCZ","download_json":"https://pith.science/pith/SMN2ZANVXUG7Q3C7UINL55VXCZ.json","view_paper":"https://pith.science/paper/SMN2ZANV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2407.09867&json=true","fetch_graph":"https://pith.science/api/pith-number/SMN2ZANVXUG7Q3C7UINL55VXCZ/graph.json","fetch_events":"https://pith.science/api/pith-number/SMN2ZANVXUG7Q3C7UINL55VXCZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SMN2ZANVXUG7Q3C7UINL55VXCZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SMN2ZANVXUG7Q3C7UINL55VXCZ/action/storage_attestation","attest_author":"https://pith.science/pith/SMN2ZANVXUG7Q3C7UINL55VXCZ/action/author_attestation","sign_citation":"https://pith.science/pith/SMN2ZANVXUG7Q3C7UINL55VXCZ/action/citation_signature","submit_replication":"https://pith.science/pith/SMN2ZANVXUG7Q3C7UINL55VXCZ/action/replication_record"}},"created_at":"2026-07-05T08:43:35.886437+00:00","updated_at":"2026-07-05T08:43:35.886437+00:00"}