{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:XGATHKYY666GY2LE2HVGC7L2RC","short_pith_number":"pith:XGATHKYY","canonical_record":{"source":{"id":"1607.00328","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-07-01T17:44:33Z","cross_cats_sorted":["math.MG","math.OA"],"title_canon_sha256":"2e9a68d0af8af53894ae427de4fe841c1d274c2b333e2bd3451210dbf38d9c68","abstract_canon_sha256":"0b25a1d3a431a3223d378b0db790213520c6ffd9359c8d63bd392a7043126eb6"},"schema_version":"1.0"},"canonical_sha256":"b98133ab18f7bc6c6964d1ea617d7a88998e6d1af2cb824495a06e7c7d964a5b","source":{"kind":"arxiv","id":"1607.00328","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.00328","created_at":"2026-05-18T00:08:48Z"},{"alias_kind":"arxiv_version","alias_value":"1607.00328v3","created_at":"2026-05-18T00:08:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.00328","created_at":"2026-05-18T00:08:48Z"},{"alias_kind":"pith_short_12","alias_value":"XGATHKYY666G","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XGATHKYY666GY2LE","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XGATHKYY","created_at":"2026-05-18T12:30:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:XGATHKYY666GY2LE2HVGC7L2RC","target":"record","payload":{"canonical_record":{"source":{"id":"1607.00328","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-07-01T17:44:33Z","cross_cats_sorted":["math.MG","math.OA"],"title_canon_sha256":"2e9a68d0af8af53894ae427de4fe841c1d274c2b333e2bd3451210dbf38d9c68","abstract_canon_sha256":"0b25a1d3a431a3223d378b0db790213520c6ffd9359c8d63bd392a7043126eb6"},"schema_version":"1.0"},"canonical_sha256":"b98133ab18f7bc6c6964d1ea617d7a88998e6d1af2cb824495a06e7c7d964a5b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:48.789617Z","signature_b64":"RIOf0QoECX8+qcPFQrhnPherAA+lJ16CsWhVuFM6NkRVRUcB5lx+RoTBNUmDZ6b3/zVc1JOBt0zyK+MaZzGQDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b98133ab18f7bc6c6964d1ea617d7a88998e6d1af2cb824495a06e7c7d964a5b","last_reissued_at":"2026-05-18T00:08:48.789134Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:48.789134Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1607.00328","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":"2026-05-18T00:08:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O+nx/ORkPZjC0UKeTNmENSzCCXpPdIiMfeta7cdGQMomRCO6X2SSL228FXt0svr4LhSstoj+N3If168iGaolAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T18:11:40.109581Z"},"content_sha256":"20948ded7ca7c788c34f292962845260c0da8477fed7c17fb19269a29e09b9ea","schema_version":"1.0","event_id":"sha256:20948ded7ca7c788c34f292962845260c0da8477fed7c17fb19269a29e09b9ea"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:XGATHKYY666GY2LE2HVGC7L2RC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Amenability of coarse spaces and K-algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.OA"],"primary_cat":"math.RA","authors_text":"Fernando Lled\\'o, Jianchao Wu, Kang Li, Pere Ara","submitted_at":"2016-07-01T17:44:33Z","abstract_excerpt":"In this article we analyze the notions of amenability and paradoxical decomposition from an algebraic perspective. We consider this dichotomy for locally finite extended metric spaces and for general algebras over commutative fields. In the context of algebras we also study the relation of amenability with proper infiniteness. We apply our general analysis to two important classes of algebras: the unital Leavitt path algebras and the translation algebras on locally finite extended metric spaces. In particular, we show that the amenability of a metric space is equivalent to the algebraic amenab"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.00328","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":"2026-05-18T00:08:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AxJmA/MWM8X4X2e3EZUE+5Uq5nrXvftla1cpcfXdasn2yt4kxmjy0Db1M9zsDbwgs7yspZg7WWt4/XJyfnNNBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T18:11:40.109936Z"},"content_sha256":"f6ffdb4fa9f44bb41fdb3d2e54ca17bb312b2fbaf9a19e7a8e56430a4b17fe47","schema_version":"1.0","event_id":"sha256:f6ffdb4fa9f44bb41fdb3d2e54ca17bb312b2fbaf9a19e7a8e56430a4b17fe47"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XGATHKYY666GY2LE2HVGC7L2RC/bundle.json","state_url":"https://pith.science/pith/XGATHKYY666GY2LE2HVGC7L2RC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XGATHKYY666GY2LE2HVGC7L2RC/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":"2026-06-04T18:11:40Z","links":{"resolver":"https://pith.science/pith/XGATHKYY666GY2LE2HVGC7L2RC","bundle":"https://pith.science/pith/XGATHKYY666GY2LE2HVGC7L2RC/bundle.json","state":"https://pith.science/pith/XGATHKYY666GY2LE2HVGC7L2RC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XGATHKYY666GY2LE2HVGC7L2RC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:XGATHKYY666GY2LE2HVGC7L2RC","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":"0b25a1d3a431a3223d378b0db790213520c6ffd9359c8d63bd392a7043126eb6","cross_cats_sorted":["math.MG","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-07-01T17:44:33Z","title_canon_sha256":"2e9a68d0af8af53894ae427de4fe841c1d274c2b333e2bd3451210dbf38d9c68"},"schema_version":"1.0","source":{"id":"1607.00328","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.00328","created_at":"2026-05-18T00:08:48Z"},{"alias_kind":"arxiv_version","alias_value":"1607.00328v3","created_at":"2026-05-18T00:08:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.00328","created_at":"2026-05-18T00:08:48Z"},{"alias_kind":"pith_short_12","alias_value":"XGATHKYY666G","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XGATHKYY666GY2LE","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XGATHKYY","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:f6ffdb4fa9f44bb41fdb3d2e54ca17bb312b2fbaf9a19e7a8e56430a4b17fe47","target":"graph","created_at":"2026-05-18T00:08:48Z","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":"In this article we analyze the notions of amenability and paradoxical decomposition from an algebraic perspective. We consider this dichotomy for locally finite extended metric spaces and for general algebras over commutative fields. In the context of algebras we also study the relation of amenability with proper infiniteness. We apply our general analysis to two important classes of algebras: the unital Leavitt path algebras and the translation algebras on locally finite extended metric spaces. In particular, we show that the amenability of a metric space is equivalent to the algebraic amenab","authors_text":"Fernando Lled\\'o, Jianchao Wu, Kang Li, Pere Ara","cross_cats":["math.MG","math.OA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-07-01T17:44:33Z","title":"Amenability of coarse spaces and K-algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.00328","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":"sha256:20948ded7ca7c788c34f292962845260c0da8477fed7c17fb19269a29e09b9ea","target":"record","created_at":"2026-05-18T00:08:48Z","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":"0b25a1d3a431a3223d378b0db790213520c6ffd9359c8d63bd392a7043126eb6","cross_cats_sorted":["math.MG","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2016-07-01T17:44:33Z","title_canon_sha256":"2e9a68d0af8af53894ae427de4fe841c1d274c2b333e2bd3451210dbf38d9c68"},"schema_version":"1.0","source":{"id":"1607.00328","kind":"arxiv","version":3}},"canonical_sha256":"b98133ab18f7bc6c6964d1ea617d7a88998e6d1af2cb824495a06e7c7d964a5b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b98133ab18f7bc6c6964d1ea617d7a88998e6d1af2cb824495a06e7c7d964a5b","first_computed_at":"2026-05-18T00:08:48.789134Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:08:48.789134Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RIOf0QoECX8+qcPFQrhnPherAA+lJ16CsWhVuFM6NkRVRUcB5lx+RoTBNUmDZ6b3/zVc1JOBt0zyK+MaZzGQDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:08:48.789617Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.00328","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:20948ded7ca7c788c34f292962845260c0da8477fed7c17fb19269a29e09b9ea","sha256:f6ffdb4fa9f44bb41fdb3d2e54ca17bb312b2fbaf9a19e7a8e56430a4b17fe47"],"state_sha256":"8e99317be43e6884737755dbb94565761f8d741fd6c19dd35e5de949544ebc1b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lV20RPUL7g5+coSJMcnXWd1IQ8ujYVdZbHaEA1X7Bo+9YTM88nyUsN1R43q6UWww04oL4rP9dhKlMgcQm9z5DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T18:11:40.111920Z","bundle_sha256":"37d38c580e4c4ac791e31d4beb885139024f4d65c66fa1e4f20dc4d9665a740e"}}