{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:ZKZWVYEGLOOYVUTQMYOVTLJ6RD","short_pith_number":"pith:ZKZWVYEG","canonical_record":{"source":{"id":"1004.3721","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-04-21T14:55:24Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"5cf58a8d30a11eb8161600b46cc3fda301e1bd1ed0d386ab7deebb73e9b8a31a","abstract_canon_sha256":"11cf3ee815f6f93af90c76258efd37c1b4596e24e5fb3d634ec5a101c072132b"},"schema_version":"1.0"},"canonical_sha256":"cab36ae0865b9d8ad270661d59ad3e88dd8394eba4738453d09628579fb5394d","source":{"kind":"arxiv","id":"1004.3721","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.3721","created_at":"2026-05-18T04:23:11Z"},{"alias_kind":"arxiv_version","alias_value":"1004.3721v3","created_at":"2026-05-18T04:23:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.3721","created_at":"2026-05-18T04:23:11Z"},{"alias_kind":"pith_short_12","alias_value":"ZKZWVYEGLOOY","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"ZKZWVYEGLOOYVUTQ","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"ZKZWVYEG","created_at":"2026-05-18T12:26:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:ZKZWVYEGLOOYVUTQMYOVTLJ6RD","target":"record","payload":{"canonical_record":{"source":{"id":"1004.3721","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-04-21T14:55:24Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"5cf58a8d30a11eb8161600b46cc3fda301e1bd1ed0d386ab7deebb73e9b8a31a","abstract_canon_sha256":"11cf3ee815f6f93af90c76258efd37c1b4596e24e5fb3d634ec5a101c072132b"},"schema_version":"1.0"},"canonical_sha256":"cab36ae0865b9d8ad270661d59ad3e88dd8394eba4738453d09628579fb5394d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:23:11.735785Z","signature_b64":"ylMV08D372ayAOBgOrzRyxQY/OOoB5S6Hr0GrlO1RKasNL8MkIwX6SekU98lOf7xLK5inJAfNJqCar4UEXEYCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cab36ae0865b9d8ad270661d59ad3e88dd8394eba4738453d09628579fb5394d","last_reissued_at":"2026-05-18T04:23:11.735367Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:23:11.735367Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1004.3721","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:23:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gnc2gLV3oR2fXiaVMblfHxYWxERYZKTnp6KgLINhIaw/uzVgNxKsP7zmXqkXb1EfBPCQjOycsY4MozHFqm+FCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T01:59:52.653927Z"},"content_sha256":"f784a085b4d3c77dc5713fda8574144b79c37e3160ca76c0c4f7275860fc6478","schema_version":"1.0","event_id":"sha256:f784a085b4d3c77dc5713fda8574144b79c37e3160ca76c0c4f7275860fc6478"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:ZKZWVYEGLOOYVUTQMYOVTLJ6RD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Reduced Amalgamated Free Products of C*-algebras and the MF-Property","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.OA","authors_text":"Jonas Andersen Seebach","submitted_at":"2010-04-21T14:55:24Z","abstract_excerpt":"We establish the MF property of the reduced group $ C^* $-algebra of an amalgamated free product of countable Abelian discrete groups. This result is then used to give a characterization of the amalgamated free products of Abelian groups for which the BDF semigroup of the reduced group $ C^* $-algebra is a group. Along the way we get a tensor product factorization of the corresponding group von Neumann algebra. We end the exposition by applying the ideas from the first part to give a few more examples of groups with a reduced group $ C^* $-algebra which is MF."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.3721","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:23:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RTlr5YXiOgouhCuHCMTQRtl235wbwF3SQQ3skr5g9xQDU+mXq5IjYJi4DDfe+ZCMkzZG2oJkB0pcr+7JYzP0Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T01:59:52.654488Z"},"content_sha256":"7a01258063ea71f4227fa00ba8357f8e81ea4d77c43974ead396ab317df59f0e","schema_version":"1.0","event_id":"sha256:7a01258063ea71f4227fa00ba8357f8e81ea4d77c43974ead396ab317df59f0e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZKZWVYEGLOOYVUTQMYOVTLJ6RD/bundle.json","state_url":"https://pith.science/pith/ZKZWVYEGLOOYVUTQMYOVTLJ6RD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZKZWVYEGLOOYVUTQMYOVTLJ6RD/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-29T01:59:52Z","links":{"resolver":"https://pith.science/pith/ZKZWVYEGLOOYVUTQMYOVTLJ6RD","bundle":"https://pith.science/pith/ZKZWVYEGLOOYVUTQMYOVTLJ6RD/bundle.json","state":"https://pith.science/pith/ZKZWVYEGLOOYVUTQMYOVTLJ6RD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZKZWVYEGLOOYVUTQMYOVTLJ6RD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:ZKZWVYEGLOOYVUTQMYOVTLJ6RD","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":"11cf3ee815f6f93af90c76258efd37c1b4596e24e5fb3d634ec5a101c072132b","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-04-21T14:55:24Z","title_canon_sha256":"5cf58a8d30a11eb8161600b46cc3fda301e1bd1ed0d386ab7deebb73e9b8a31a"},"schema_version":"1.0","source":{"id":"1004.3721","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.3721","created_at":"2026-05-18T04:23:11Z"},{"alias_kind":"arxiv_version","alias_value":"1004.3721v3","created_at":"2026-05-18T04:23:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.3721","created_at":"2026-05-18T04:23:11Z"},{"alias_kind":"pith_short_12","alias_value":"ZKZWVYEGLOOY","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"ZKZWVYEGLOOYVUTQ","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"ZKZWVYEG","created_at":"2026-05-18T12:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:7a01258063ea71f4227fa00ba8357f8e81ea4d77c43974ead396ab317df59f0e","target":"graph","created_at":"2026-05-18T04:23:11Z","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 establish the MF property of the reduced group $ C^* $-algebra of an amalgamated free product of countable Abelian discrete groups. This result is then used to give a characterization of the amalgamated free products of Abelian groups for which the BDF semigroup of the reduced group $ C^* $-algebra is a group. Along the way we get a tensor product factorization of the corresponding group von Neumann algebra. We end the exposition by applying the ideas from the first part to give a few more examples of groups with a reduced group $ C^* $-algebra which is MF.","authors_text":"Jonas Andersen Seebach","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-04-21T14:55:24Z","title":"On Reduced Amalgamated Free Products of C*-algebras and the MF-Property"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.3721","kind":"arxiv","version":3},"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:f784a085b4d3c77dc5713fda8574144b79c37e3160ca76c0c4f7275860fc6478","target":"record","created_at":"2026-05-18T04:23:11Z","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":"11cf3ee815f6f93af90c76258efd37c1b4596e24e5fb3d634ec5a101c072132b","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2010-04-21T14:55:24Z","title_canon_sha256":"5cf58a8d30a11eb8161600b46cc3fda301e1bd1ed0d386ab7deebb73e9b8a31a"},"schema_version":"1.0","source":{"id":"1004.3721","kind":"arxiv","version":3}},"canonical_sha256":"cab36ae0865b9d8ad270661d59ad3e88dd8394eba4738453d09628579fb5394d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cab36ae0865b9d8ad270661d59ad3e88dd8394eba4738453d09628579fb5394d","first_computed_at":"2026-05-18T04:23:11.735367Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:23:11.735367Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ylMV08D372ayAOBgOrzRyxQY/OOoB5S6Hr0GrlO1RKasNL8MkIwX6SekU98lOf7xLK5inJAfNJqCar4UEXEYCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:23:11.735785Z","signed_message":"canonical_sha256_bytes"},"source_id":"1004.3721","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f784a085b4d3c77dc5713fda8574144b79c37e3160ca76c0c4f7275860fc6478","sha256:7a01258063ea71f4227fa00ba8357f8e81ea4d77c43974ead396ab317df59f0e"],"state_sha256":"b8805c3f837c37fad2e45c3af0865dd377a05f76ade5cc2ef502a542726f7762"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"N+vW+NG9wnjwZznZ84Jf9e+fAoHL6hRmH15wn9ydYT9aCT3xvi2JDQumqESJwaukfgojQHLgoFlPO81GiT+jBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T01:59:52.657011Z","bundle_sha256":"91fae1349e431acab22fd9ba8d0a5e83b0d1ab45d42d7e7ce69d61f8b4522b9f"}}