{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:TK7K4YJP27LYV3CNTFMQF4FPN4","short_pith_number":"pith:TK7K4YJP","canonical_record":{"source":{"id":"1511.05312","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2015-11-17T08:55:37Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"a968b0ef6791b1d56176aa0e0babab49d8abfa0040f189fdd9761d77cbd3857a","abstract_canon_sha256":"832c485c217270d7dea27dd21ab6d50c8c7b94d40ebf04becacd54daf3ae2fc1"},"schema_version":"1.0"},"canonical_sha256":"9abeae612fd7d78aec4d995902f0af6f2150597296ddc199798c4b06fe2ea00f","source":{"kind":"arxiv","id":"1511.05312","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.05312","created_at":"2026-05-18T01:21:04Z"},{"alias_kind":"arxiv_version","alias_value":"1511.05312v3","created_at":"2026-05-18T01:21:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.05312","created_at":"2026-05-18T01:21:04Z"},{"alias_kind":"pith_short_12","alias_value":"TK7K4YJP27LY","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"TK7K4YJP27LYV3CN","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"TK7K4YJP","created_at":"2026-05-18T12:29:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:TK7K4YJP27LYV3CNTFMQF4FPN4","target":"record","payload":{"canonical_record":{"source":{"id":"1511.05312","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2015-11-17T08:55:37Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"a968b0ef6791b1d56176aa0e0babab49d8abfa0040f189fdd9761d77cbd3857a","abstract_canon_sha256":"832c485c217270d7dea27dd21ab6d50c8c7b94d40ebf04becacd54daf3ae2fc1"},"schema_version":"1.0"},"canonical_sha256":"9abeae612fd7d78aec4d995902f0af6f2150597296ddc199798c4b06fe2ea00f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:04.144728Z","signature_b64":"jlg0T3bpYn73KPPFVNDr09NZVrk79fSkrSyje26fCIELYpZBv42a5tiMelgxXeYR0Yx07/9QVxMXUwG6o4jODg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9abeae612fd7d78aec4d995902f0af6f2150597296ddc199798c4b06fe2ea00f","last_reissued_at":"2026-05-18T01:21:04.144186Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:04.144186Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1511.05312","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":"2026-05-18T01:21:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xR6zBXEmPy31oYBmateLQmgHk7iMlqtgV7kULOvSlclTpv9fmxDD3kvmPdKcTp0kyBnznZVpzFYx0N3yuZJsDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T23:37:04.535258Z"},"content_sha256":"dda14ddd1aaff1a1a8a4f740a3dfb84392392a3811d35c9f5a636cb8d59cf2a5","schema_version":"1.0","event_id":"sha256:dda14ddd1aaff1a1a8a4f740a3dfb84392392a3811d35c9f5a636cb8d59cf2a5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:TK7K4YJP27LYV3CNTFMQF4FPN4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Notes on twisted equivariant $\\mathrm{K}$-theory for $\\mathrm{C}^*$-algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.KT","authors_text":"Yosuke Kubota","submitted_at":"2015-11-17T08:55:37Z","abstract_excerpt":"In this paper, we study a generalization of twisted (groupoid) equivariant $\\mathrm{K}$-theory in the sense of Freed-Moore for $\\mathbb{Z}_2$-graded $\\mathrm{C}^*$-algebras. It is defined by using Fredholm operators on Hilbert modules with twisted representations. We compare it with another description using odd symmetries, which is a generalization of van Daele's $\\mathrm{K}$-theory for $\\mathbb{Z}_2$-graded Banach algebras. In particular, we obtain a simple presentation of the twisted equivariant $\\mathrm{K}$-group when the $\\mathrm{C}^*$-algebra is trivially graded. It is applied for the bu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.05312","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":"2026-05-18T01:21:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NjxNNX5KRje4CAL2Hh9M89qm54noHbJN074ptFG96KKuDJziR6aJ3DzSWcc+2Og1f0gGRoOgWnigkHH23CBeDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T23:37:04.535623Z"},"content_sha256":"ed5f68729c31973c796ec3ccd2a2a974ecb38e75a22c85a1b6743ee48ba3cf4e","schema_version":"1.0","event_id":"sha256:ed5f68729c31973c796ec3ccd2a2a974ecb38e75a22c85a1b6743ee48ba3cf4e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TK7K4YJP27LYV3CNTFMQF4FPN4/bundle.json","state_url":"https://pith.science/pith/TK7K4YJP27LYV3CNTFMQF4FPN4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TK7K4YJP27LYV3CNTFMQF4FPN4/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":"2026-06-08T23:37:04Z","links":{"resolver":"https://pith.science/pith/TK7K4YJP27LYV3CNTFMQF4FPN4","bundle":"https://pith.science/pith/TK7K4YJP27LYV3CNTFMQF4FPN4/bundle.json","state":"https://pith.science/pith/TK7K4YJP27LYV3CNTFMQF4FPN4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TK7K4YJP27LYV3CNTFMQF4FPN4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:TK7K4YJP27LYV3CNTFMQF4FPN4","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":"832c485c217270d7dea27dd21ab6d50c8c7b94d40ebf04becacd54daf3ae2fc1","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2015-11-17T08:55:37Z","title_canon_sha256":"a968b0ef6791b1d56176aa0e0babab49d8abfa0040f189fdd9761d77cbd3857a"},"schema_version":"1.0","source":{"id":"1511.05312","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.05312","created_at":"2026-05-18T01:21:04Z"},{"alias_kind":"arxiv_version","alias_value":"1511.05312v3","created_at":"2026-05-18T01:21:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.05312","created_at":"2026-05-18T01:21:04Z"},{"alias_kind":"pith_short_12","alias_value":"TK7K4YJP27LY","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"TK7K4YJP27LYV3CN","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"TK7K4YJP","created_at":"2026-05-18T12:29:42Z"}],"graph_snapshots":[{"event_id":"sha256:ed5f68729c31973c796ec3ccd2a2a974ecb38e75a22c85a1b6743ee48ba3cf4e","target":"graph","created_at":"2026-05-18T01:21:04Z","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 paper, we study a generalization of twisted (groupoid) equivariant $\\mathrm{K}$-theory in the sense of Freed-Moore for $\\mathbb{Z}_2$-graded $\\mathrm{C}^*$-algebras. It is defined by using Fredholm operators on Hilbert modules with twisted representations. We compare it with another description using odd symmetries, which is a generalization of van Daele's $\\mathrm{K}$-theory for $\\mathbb{Z}_2$-graded Banach algebras. In particular, we obtain a simple presentation of the twisted equivariant $\\mathrm{K}$-group when the $\\mathrm{C}^*$-algebra is trivially graded. It is applied for the bu","authors_text":"Yosuke Kubota","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2015-11-17T08:55:37Z","title":"Notes on twisted equivariant $\\mathrm{K}$-theory for $\\mathrm{C}^*$-algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.05312","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:dda14ddd1aaff1a1a8a4f740a3dfb84392392a3811d35c9f5a636cb8d59cf2a5","target":"record","created_at":"2026-05-18T01:21:04Z","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":"832c485c217270d7dea27dd21ab6d50c8c7b94d40ebf04becacd54daf3ae2fc1","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2015-11-17T08:55:37Z","title_canon_sha256":"a968b0ef6791b1d56176aa0e0babab49d8abfa0040f189fdd9761d77cbd3857a"},"schema_version":"1.0","source":{"id":"1511.05312","kind":"arxiv","version":3}},"canonical_sha256":"9abeae612fd7d78aec4d995902f0af6f2150597296ddc199798c4b06fe2ea00f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9abeae612fd7d78aec4d995902f0af6f2150597296ddc199798c4b06fe2ea00f","first_computed_at":"2026-05-18T01:21:04.144186Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:21:04.144186Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jlg0T3bpYn73KPPFVNDr09NZVrk79fSkrSyje26fCIELYpZBv42a5tiMelgxXeYR0Yx07/9QVxMXUwG6o4jODg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:21:04.144728Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.05312","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dda14ddd1aaff1a1a8a4f740a3dfb84392392a3811d35c9f5a636cb8d59cf2a5","sha256:ed5f68729c31973c796ec3ccd2a2a974ecb38e75a22c85a1b6743ee48ba3cf4e"],"state_sha256":"326d4797fcb1fa0d24ea09d4f74b998064f4cffaee65f720764f4e3e6130d01f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ky1AtelxzJNfvuUu+gtAB1tKFn1K+JOT4iFDkdLKrArvjRrh82S4Y8LKb/E+Rq/4NryEdchCSoYgVWbvSitWAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T23:37:04.537476Z","bundle_sha256":"8e87beab24d236e2f19f8fe2fd5f504aa00b224d51efbb670ed9eb1775da619f"}}