{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:XT6R2LPNATV3N3CUX3YZZYHTJW","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":"47ea7fe3c272394e05c571d1932a88b747b344f91db9a81f953f0f39bc8ba60b","cross_cats_sorted":["math.GR"],"license":"","primary_cat":"math.OA","submitted_at":"2005-05-27T18:41:05Z","title_canon_sha256":"6da3c8ca5eea036f12391ddd9c60915d6269ec893d495e4b4664edf19704bdd3"},"schema_version":"1.0","source":{"id":"math/0505589","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0505589","created_at":"2026-07-04T15:06:29Z"},{"alias_kind":"arxiv_version","alias_value":"math/0505589v6","created_at":"2026-07-04T15:06:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0505589","created_at":"2026-07-04T15:06:29Z"},{"alias_kind":"pith_short_12","alias_value":"XT6R2LPNATV3","created_at":"2026-07-04T15:06:29Z"},{"alias_kind":"pith_short_16","alias_value":"XT6R2LPNATV3N3CU","created_at":"2026-07-04T15:06:29Z"},{"alias_kind":"pith_short_8","alias_value":"XT6R2LPN","created_at":"2026-07-04T15:06:29Z"}],"graph_snapshots":[{"event_id":"sha256:5dfcf92400e30e451e27fd948c468de5c79028a65566d9e7223c45d932484c92","target":"graph","created_at":"2026-07-04T15:06:29Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/math/0505589/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider amalgamated free product II$_1$ factors $M = M_1 *_B M_2 *_B ...$ and use ``deformation/rigidity'' and ``intertwining'' techniques to prove that any relatively rigid von Neumann subalgebra $Q\\subset M$ can be intertwined into one of the $M_i$'s. We apply this to the case $M_i$ are w-rigid II$_1$ factors, with $B$ equal to either $\\Bbb C$, to a Cartan subalgebra $A$ in $M_i$, or to a regular hyperfinite II$_1$ subfactor $R$ in $M_i$, to obtain the following type of unique decomposition results, \\`a la Bass-Serre: If $M = (N_1 *_C N_2 *_C ...)^t$, for some $t>0$ and some other simila","authors_text":"A. Ioana, J. Peterson, S. Popa","cross_cats":["math.GR"],"headline":"","license":"","primary_cat":"math.OA","submitted_at":"2005-05-27T18:41:05Z","title":"Amalgamated Free Products of $w$-Rigid Factors and Calculation of their Symmetry Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0505589","kind":"arxiv","version":6},"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:fc9c955b2e554e4f13676855f39d4d593c97086c3c121f36c58853c0c99200aa","target":"record","created_at":"2026-07-04T15:06:29Z","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":"47ea7fe3c272394e05c571d1932a88b747b344f91db9a81f953f0f39bc8ba60b","cross_cats_sorted":["math.GR"],"license":"","primary_cat":"math.OA","submitted_at":"2005-05-27T18:41:05Z","title_canon_sha256":"6da3c8ca5eea036f12391ddd9c60915d6269ec893d495e4b4664edf19704bdd3"},"schema_version":"1.0","source":{"id":"math/0505589","kind":"arxiv","version":6}},"canonical_sha256":"bcfd1d2ded04ebb6ec54bef19ce0f34da581e49d54855ead835c6b9bad99a787","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bcfd1d2ded04ebb6ec54bef19ce0f34da581e49d54855ead835c6b9bad99a787","first_computed_at":"2026-07-04T15:06:29.995692Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:06:29.995692Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QvgAakhmK6Y4PY62VocFgMqY4k8Bj4RlNm24ay5Lia/edoNdEevL5HaI+lYKE3hmCWL2bkQ1FHxbECLUKSupAA==","signature_status":"signed_v1","signed_at":"2026-07-04T15:06:29.996100Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0505589","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fc9c955b2e554e4f13676855f39d4d593c97086c3c121f36c58853c0c99200aa","sha256:5dfcf92400e30e451e27fd948c468de5c79028a65566d9e7223c45d932484c92"],"state_sha256":"78628609935ef50f34bdf95d4ce0689950040a7ac7b653f4f4eb0049b2cd6aa3"}