{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:6WFJZN37272GGZHU3TJ6JBJOF7","short_pith_number":"pith:6WFJZN37","canonical_record":{"source":{"id":"1505.06633","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2015-05-25T14:09:04Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"5fee06e80424e0eaafa2802576179874c44977f78993647d06d64f92f4dcf3e9","abstract_canon_sha256":"ed64aeff1a3464c7455aecdd162a9aaed64ae0e516cd2bc7e6d64bf1e2cb959f"},"schema_version":"1.0"},"canonical_sha256":"f58a9cb77fd7f46364f4dcd3e4852e2fd968b28b8092392c69c94a5b4ac1cb55","source":{"kind":"arxiv","id":"1505.06633","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.06633","created_at":"2026-05-18T01:19:53Z"},{"alias_kind":"arxiv_version","alias_value":"1505.06633v3","created_at":"2026-05-18T01:19:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.06633","created_at":"2026-05-18T01:19:53Z"},{"alias_kind":"pith_short_12","alias_value":"6WFJZN37272G","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6WFJZN37272GGZHU","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6WFJZN37","created_at":"2026-05-18T12:29:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:6WFJZN37272GGZHU3TJ6JBJOF7","target":"record","payload":{"canonical_record":{"source":{"id":"1505.06633","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2015-05-25T14:09:04Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"5fee06e80424e0eaafa2802576179874c44977f78993647d06d64f92f4dcf3e9","abstract_canon_sha256":"ed64aeff1a3464c7455aecdd162a9aaed64ae0e516cd2bc7e6d64bf1e2cb959f"},"schema_version":"1.0"},"canonical_sha256":"f58a9cb77fd7f46364f4dcd3e4852e2fd968b28b8092392c69c94a5b4ac1cb55","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:53.858138Z","signature_b64":"DPEqDTa8GJr6qn0+PoZ6IOfaiRANplUTZsZP6LBQglpRQzL7HYv01+2fE3nkYYo0n3I2YV8vu/1yvuDX8wOfCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f58a9cb77fd7f46364f4dcd3e4852e2fd968b28b8092392c69c94a5b4ac1cb55","last_reissued_at":"2026-05-18T01:19:53.857663Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:53.857663Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1505.06633","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":"2026-05-18T01:19:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UXd/z370k3cT/zrg3O/83zo36LYAcEchjzSSqEo9aXGF51fT/SYhJZ4/znjtI4P6lwxB/NdRiWloB5N/ciSYBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T13:26:03.758992Z"},"content_sha256":"ea2baf92b5cebb092e46f2a923c44dc08ff7c5d035e3a8413f98151955187558","schema_version":"1.0","event_id":"sha256:ea2baf92b5cebb092e46f2a923c44dc08ff7c5d035e3a8413f98151955187558"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:6WFJZN37272GGZHU3TJ6JBJOF7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ergodic invariant states and irreducible representations of crossed product $C^*$-algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.OA","authors_text":"Huichi Huang, Jianchao Wu","submitted_at":"2015-05-25T14:09:04Z","abstract_excerpt":"Motivated by reformulating Furstenberg's $\\times p,\\times q$ conjecture via representations of a crossed product $C^*$-algebra, we show that in a discrete $C^*$-dynamical system $(A,\\Gamma)$, the space of (ergodic) $\\Gamma$-invariant states on $A$ is homeomorphic to a subspace of (pure) state space of $A\\rtimes\\Gamma$. Various applications of this in topological dynamical systems and representation theory are obtained. In particular, we prove that the classification of ergodic $\\Gamma$-invariant regular Borel probability measures on a compact Hausdorff space $X$ is equivalent to the classifica"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.06633","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":"2026-05-18T01:19:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"P2HYWSDdB+xqBQWaI/2SCfLLcifMMBvjix31hspk+AcYQU6CorDAC6Ea2wKyNyZS7p7trQxulqxSgxMbt2coBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T13:26:03.759353Z"},"content_sha256":"29dab95fcea8ad4238bf17d7fc3e9174189fa52bd5122d9dbc3f84d95fbbaa3f","schema_version":"1.0","event_id":"sha256:29dab95fcea8ad4238bf17d7fc3e9174189fa52bd5122d9dbc3f84d95fbbaa3f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6WFJZN37272GGZHU3TJ6JBJOF7/bundle.json","state_url":"https://pith.science/pith/6WFJZN37272GGZHU3TJ6JBJOF7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6WFJZN37272GGZHU3TJ6JBJOF7/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":"2026-06-24T13:26:03Z","links":{"resolver":"https://pith.science/pith/6WFJZN37272GGZHU3TJ6JBJOF7","bundle":"https://pith.science/pith/6WFJZN37272GGZHU3TJ6JBJOF7/bundle.json","state":"https://pith.science/pith/6WFJZN37272GGZHU3TJ6JBJOF7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6WFJZN37272GGZHU3TJ6JBJOF7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:6WFJZN37272GGZHU3TJ6JBJOF7","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":"ed64aeff1a3464c7455aecdd162a9aaed64ae0e516cd2bc7e6d64bf1e2cb959f","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2015-05-25T14:09:04Z","title_canon_sha256":"5fee06e80424e0eaafa2802576179874c44977f78993647d06d64f92f4dcf3e9"},"schema_version":"1.0","source":{"id":"1505.06633","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.06633","created_at":"2026-05-18T01:19:53Z"},{"alias_kind":"arxiv_version","alias_value":"1505.06633v3","created_at":"2026-05-18T01:19:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.06633","created_at":"2026-05-18T01:19:53Z"},{"alias_kind":"pith_short_12","alias_value":"6WFJZN37272G","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6WFJZN37272GGZHU","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6WFJZN37","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:29dab95fcea8ad4238bf17d7fc3e9174189fa52bd5122d9dbc3f84d95fbbaa3f","target":"graph","created_at":"2026-05-18T01:19:53Z","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":"Motivated by reformulating Furstenberg's $\\times p,\\times q$ conjecture via representations of a crossed product $C^*$-algebra, we show that in a discrete $C^*$-dynamical system $(A,\\Gamma)$, the space of (ergodic) $\\Gamma$-invariant states on $A$ is homeomorphic to a subspace of (pure) state space of $A\\rtimes\\Gamma$. Various applications of this in topological dynamical systems and representation theory are obtained. In particular, we prove that the classification of ergodic $\\Gamma$-invariant regular Borel probability measures on a compact Hausdorff space $X$ is equivalent to the classifica","authors_text":"Huichi Huang, Jianchao Wu","cross_cats":["math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2015-05-25T14:09:04Z","title":"Ergodic invariant states and irreducible representations of crossed product $C^*$-algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.06633","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:ea2baf92b5cebb092e46f2a923c44dc08ff7c5d035e3a8413f98151955187558","target":"record","created_at":"2026-05-18T01:19:53Z","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":"ed64aeff1a3464c7455aecdd162a9aaed64ae0e516cd2bc7e6d64bf1e2cb959f","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2015-05-25T14:09:04Z","title_canon_sha256":"5fee06e80424e0eaafa2802576179874c44977f78993647d06d64f92f4dcf3e9"},"schema_version":"1.0","source":{"id":"1505.06633","kind":"arxiv","version":3}},"canonical_sha256":"f58a9cb77fd7f46364f4dcd3e4852e2fd968b28b8092392c69c94a5b4ac1cb55","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f58a9cb77fd7f46364f4dcd3e4852e2fd968b28b8092392c69c94a5b4ac1cb55","first_computed_at":"2026-05-18T01:19:53.857663Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:53.857663Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DPEqDTa8GJr6qn0+PoZ6IOfaiRANplUTZsZP6LBQglpRQzL7HYv01+2fE3nkYYo0n3I2YV8vu/1yvuDX8wOfCA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:53.858138Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.06633","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ea2baf92b5cebb092e46f2a923c44dc08ff7c5d035e3a8413f98151955187558","sha256:29dab95fcea8ad4238bf17d7fc3e9174189fa52bd5122d9dbc3f84d95fbbaa3f"],"state_sha256":"b602500de2b3780c0da3ffe4c971afdaf0dda4252772c54b9c9988227bc7b5f0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+WEHpKHyHZUlyi1HDF+GR5BtrI18KAkoyF5CvznbiZZhJh9wu7vQCNalLDj80l5fEsL7xBC93tWx7ughSHl3DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T13:26:03.761308Z","bundle_sha256":"eb1cdff91d3402263d83813b3aff1c2379564b276186daa3cd34bbcf7bf4829c"}}