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In this paper, we prove that $cel_{CU}(A)=2\\pi$ provided that $A$ is an $AH$ algebras with slow dimension growth whose real rank is not zero. On the other hand, we prove that $cel_{CU}(A)\\leq 2\\pi$ when $A$ is an $AH$ algebra with ideal property and of no dimension growth (if we further assume $A$ is not of real rank zero, we have $cel_{CU}(A)= 2\\pi$)."},"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":"1807.09018","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2018-07-24T10:26:53Z","cross_cats_sorted":[],"title_canon_sha256":"532fb1403d08585088d284a703bc83c41936b0f60fc9285231ecf19f5f194e25","abstract_canon_sha256":"8a4d39a5dbf5f124d2aa6fac04c8f9b02007b1c72e37419621aab2a1c8bbd48f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:57.017577Z","signature_b64":"ztVcoqePg5vBVOZoOXhxfMp48cdOIEmtYhMmxRaDI+QNcXtPQVaAW0xXNVBHLMuVEE5VTCB4hAJEPrjGYKlEBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"051a7c41f198e778c773c790436652a7ea8915b95f2fcde8636e92ca8191c146","last_reissued_at":"2026-05-18T00:09:57.017029Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:57.017029Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$C^*$ exponential length of commutators unitaries in $AH$ algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Chun Guang Li, Iv\\'an Vel\\'azquez Ruiz, Liangqing Li","submitted_at":"2018-07-24T10:26:53Z","abstract_excerpt":"For each unital $C^*$-algebra $A$, we denote $cel_{CU}(A)=\\sup\\{cel(u):u\\in CU(A)\\}$, where $cel(u)$ is the exponential length of $u$ and $CU(A)$ is the closure of the commutator subgroup of $U_0(A)$. In this paper, we prove that $cel_{CU}(A)=2\\pi$ provided that $A$ is an $AH$ algebras with slow dimension growth whose real rank is not zero. On the other hand, we prove that $cel_{CU}(A)\\leq 2\\pi$ when $A$ is an $AH$ algebra with ideal property and of no dimension growth (if we further assume $A$ is not of real rank zero, we have $cel_{CU}(A)= 2\\pi$)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.09018","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":""},"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":"1807.09018","created_at":"2026-05-18T00:09:57.017129+00:00"},{"alias_kind":"arxiv_version","alias_value":"1807.09018v1","created_at":"2026-05-18T00:09:57.017129+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.09018","created_at":"2026-05-18T00:09:57.017129+00:00"},{"alias_kind":"pith_short_12","alias_value":"AUNHYQPRTDTX","created_at":"2026-05-18T12:32:13.499390+00:00"},{"alias_kind":"pith_short_16","alias_value":"AUNHYQPRTDTXRR3T","created_at":"2026-05-18T12:32:13.499390+00:00"},{"alias_kind":"pith_short_8","alias_value":"AUNHYQPR","created_at":"2026-05-18T12:32:13.499390+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/AUNHYQPRTDTXRR3TY6IEGZSSU7","json":"https://pith.science/pith/AUNHYQPRTDTXRR3TY6IEGZSSU7.json","graph_json":"https://pith.science/api/pith-number/AUNHYQPRTDTXRR3TY6IEGZSSU7/graph.json","events_json":"https://pith.science/api/pith-number/AUNHYQPRTDTXRR3TY6IEGZSSU7/events.json","paper":"https://pith.science/paper/AUNHYQPR"},"agent_actions":{"view_html":"https://pith.science/pith/AUNHYQPRTDTXRR3TY6IEGZSSU7","download_json":"https://pith.science/pith/AUNHYQPRTDTXRR3TY6IEGZSSU7.json","view_paper":"https://pith.science/paper/AUNHYQPR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1807.09018&json=true","fetch_graph":"https://pith.science/api/pith-number/AUNHYQPRTDTXRR3TY6IEGZSSU7/graph.json","fetch_events":"https://pith.science/api/pith-number/AUNHYQPRTDTXRR3TY6IEGZSSU7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AUNHYQPRTDTXRR3TY6IEGZSSU7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AUNHYQPRTDTXRR3TY6IEGZSSU7/action/storage_attestation","attest_author":"https://pith.science/pith/AUNHYQPRTDTXRR3TY6IEGZSSU7/action/author_attestation","sign_citation":"https://pith.science/pith/AUNHYQPRTDTXRR3TY6IEGZSSU7/action/citation_signature","submit_replication":"https://pith.science/pith/AUNHYQPRTDTXRR3TY6IEGZSSU7/action/replication_record"}},"created_at":"2026-05-18T00:09:57.017129+00:00","updated_at":"2026-05-18T00:09:57.017129+00:00"}