{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:T3U2GXIRVLX3BRKXZGLWWEFSEE","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"7e7125822f557acc1d0939575ccd7212bdc7e4cf845474f11826657e55da64f5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-06-29T03:54:50Z","title_canon_sha256":"918af404cf16dcd968db2b77ef6fb146ae24eab9f539c630abd63ac3d4710bb3"},"schema_version":"1.0","source":{"id":"1006.5501","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1006.5501","created_at":"2026-05-18T04:41:44Z"},{"alias_kind":"arxiv_version","alias_value":"1006.5501v2","created_at":"2026-05-18T04:41:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.5501","created_at":"2026-05-18T04:41:44Z"},{"alias_kind":"pith_short_12","alias_value":"T3U2GXIRVLX3","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"T3U2GXIRVLX3BRKX","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"T3U2GXIR","created_at":"2026-05-18T12:26:13Z"}],"graph_snapshots":[{"event_id":"sha256:523ab35d35702a64a0baca8b452026f5eb6890939a5477fd625cf549b5069da6","target":"graph","created_at":"2026-05-18T04:41:44Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We introduce the fundamental group F(A) of a simple $\\sigma$-unital $C^*$-algebra $A$ with unique (up to scalar multiple) densely defined lower semicontinuous trace. This is a generalization of our previous works. Our definition in this paper makes sense for stably projectionless $C^*$-algebras. We show that there exist separable stably projectionless $C^*$-algebras such that their fundamental groups are equal to $\\mathbb{R}_+^\\times$ by using the classification theorem of Razak and Tsang. This is a contrast to the unital case. This study is motivated by the work of Kishimoto and Kumjian.","authors_text":"Norio Nawata","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-06-29T03:54:50Z","title":"Fundamental group of simple $C^*$-algebras with unique trace III"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.5501","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:74609dc0d71b6a6e4f559171de7eda3c3f039ddde0dd17bf836e186313c8f18a","target":"record","created_at":"2026-05-18T04:41:44Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"7e7125822f557acc1d0939575ccd7212bdc7e4cf845474f11826657e55da64f5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-06-29T03:54:50Z","title_canon_sha256":"918af404cf16dcd968db2b77ef6fb146ae24eab9f539c630abd63ac3d4710bb3"},"schema_version":"1.0","source":{"id":"1006.5501","kind":"arxiv","version":2}},"canonical_sha256":"9ee9a35d11aaefb0c557c9976b10b2211da46c86403e43d4921bdb49dda80681","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9ee9a35d11aaefb0c557c9976b10b2211da46c86403e43d4921bdb49dda80681","first_computed_at":"2026-05-18T04:41:44.732619Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:41:44.732619Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8Hpntq9wIpVpRntBuHcmAKt8sGMcVLrsfa5G/fm38y/tv11JkXlq90d++EugxXaOYjffvGIBv9NgBPKrE9zACg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:41:44.733118Z","signed_message":"canonical_sha256_bytes"},"source_id":"1006.5501","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:74609dc0d71b6a6e4f559171de7eda3c3f039ddde0dd17bf836e186313c8f18a","sha256:523ab35d35702a64a0baca8b452026f5eb6890939a5477fd625cf549b5069da6"],"state_sha256":"90658dd6d5f6f706b0070ad3ee8939968006dcf2e2f8a878f5bcde66062d32d3"}