{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:TDIC3WVVHHSZVL2LBYOQC625VP","short_pith_number":"pith:TDIC3WVV","canonical_record":{"source":{"id":"1611.08790","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2016-11-27T05:09:08Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"41102c4491dde678cc0d62001e276a104e55f8350381ebd188657a99c5bef02d","abstract_canon_sha256":"e9da5232738b0a61714ba09753276e75f6fd2a606811791f24562b0a9992c2e5"},"schema_version":"1.0"},"canonical_sha256":"98d02ddab539e59aaf4b0e1d017b5dabddea6b0619c1ef709dc7c2d18691751f","source":{"kind":"arxiv","id":"1611.08790","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.08790","created_at":"2026-05-18T00:19:53Z"},{"alias_kind":"arxiv_version","alias_value":"1611.08790v2","created_at":"2026-05-18T00:19:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.08790","created_at":"2026-05-18T00:19:53Z"},{"alias_kind":"pith_short_12","alias_value":"TDIC3WVVHHSZ","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_16","alias_value":"TDIC3WVVHHSZVL2L","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_8","alias_value":"TDIC3WVV","created_at":"2026-05-18T12:30:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:TDIC3WVVHHSZVL2LBYOQC625VP","target":"record","payload":{"canonical_record":{"source":{"id":"1611.08790","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2016-11-27T05:09:08Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"41102c4491dde678cc0d62001e276a104e55f8350381ebd188657a99c5bef02d","abstract_canon_sha256":"e9da5232738b0a61714ba09753276e75f6fd2a606811791f24562b0a9992c2e5"},"schema_version":"1.0"},"canonical_sha256":"98d02ddab539e59aaf4b0e1d017b5dabddea6b0619c1ef709dc7c2d18691751f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:53.840446Z","signature_b64":"HHbo7vHZRHnEzHU7yc0xMHbv10IrRTcCXMhpqCXja2ZYnqVWCV+BovbdToJh1yxLAJEwlFwmULKV9Pce4JEuBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"98d02ddab539e59aaf4b0e1d017b5dabddea6b0619c1ef709dc7c2d18691751f","last_reissued_at":"2026-05-18T00:19:53.839738Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:53.839738Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.08790","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:19:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CwzE8j+rTLZOUTkL6smvvgGuGR0zq7gMzFrTZsRh20k10bfi/+zk8oKBxiVKeOUSk/Y4hmYWjvOMZNJ/xOKWBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T14:32:42.050993Z"},"content_sha256":"492e52327be1e6078b82be70dd9c6aaebacc7a79c93c9596d284dcdd358edd4e","schema_version":"1.0","event_id":"sha256:492e52327be1e6078b82be70dd9c6aaebacc7a79c93c9596d284dcdd358edd4e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:TDIC3WVVHHSZVL2LBYOQC625VP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Quantitative $K$-theory for Banach algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.KT","authors_text":"Yeong Chyuan Chung","submitted_at":"2016-11-27T05:09:08Z","abstract_excerpt":"Quantitative (or controlled) $K$-theory for $C^*$-algebras was used by Guoliang Yu in his work on the Novikov conjecture, and later developed more formally by Yu together with Herv\\'e Oyono-Oyono. In this paper, we extend their work by developing a framework of quantitative $K$-theory for the class of algebras of bounded linear operators on subquotients (i.e., subspaces of quotients) of $L_p$ spaces. We also prove the existence of a controlled Mayer-Vietoris sequence in this framework."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.08790","kind":"arxiv","version":2},"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":"2026-05-18T00:19:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Hl+aILUMeUByi2pex8Qb+ILDa8eQuAFpaQOvpXtnXwto2YG5biOCSOkbEuila82k9ihzIKGnO3RyLCj0ZF2aDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T14:32:42.051416Z"},"content_sha256":"e6edd0f2a74d28194ca791ad7caac7064742db9c0d663af19286fe8d1c9a5151","schema_version":"1.0","event_id":"sha256:e6edd0f2a74d28194ca791ad7caac7064742db9c0d663af19286fe8d1c9a5151"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TDIC3WVVHHSZVL2LBYOQC625VP/bundle.json","state_url":"https://pith.science/pith/TDIC3WVVHHSZVL2LBYOQC625VP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TDIC3WVVHHSZVL2LBYOQC625VP/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":"2026-06-28T14:32:42Z","links":{"resolver":"https://pith.science/pith/TDIC3WVVHHSZVL2LBYOQC625VP","bundle":"https://pith.science/pith/TDIC3WVVHHSZVL2LBYOQC625VP/bundle.json","state":"https://pith.science/pith/TDIC3WVVHHSZVL2LBYOQC625VP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TDIC3WVVHHSZVL2LBYOQC625VP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:TDIC3WVVHHSZVL2LBYOQC625VP","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":"e9da5232738b0a61714ba09753276e75f6fd2a606811791f24562b0a9992c2e5","cross_cats_sorted":["math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2016-11-27T05:09:08Z","title_canon_sha256":"41102c4491dde678cc0d62001e276a104e55f8350381ebd188657a99c5bef02d"},"schema_version":"1.0","source":{"id":"1611.08790","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.08790","created_at":"2026-05-18T00:19:53Z"},{"alias_kind":"arxiv_version","alias_value":"1611.08790v2","created_at":"2026-05-18T00:19:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.08790","created_at":"2026-05-18T00:19:53Z"},{"alias_kind":"pith_short_12","alias_value":"TDIC3WVVHHSZ","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_16","alias_value":"TDIC3WVVHHSZVL2L","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_8","alias_value":"TDIC3WVV","created_at":"2026-05-18T12:30:44Z"}],"graph_snapshots":[{"event_id":"sha256:e6edd0f2a74d28194ca791ad7caac7064742db9c0d663af19286fe8d1c9a5151","target":"graph","created_at":"2026-05-18T00:19:53Z","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":"Quantitative (or controlled) $K$-theory for $C^*$-algebras was used by Guoliang Yu in his work on the Novikov conjecture, and later developed more formally by Yu together with Herv\\'e Oyono-Oyono. In this paper, we extend their work by developing a framework of quantitative $K$-theory for the class of algebras of bounded linear operators on subquotients (i.e., subspaces of quotients) of $L_p$ spaces. We also prove the existence of a controlled Mayer-Vietoris sequence in this framework.","authors_text":"Yeong Chyuan Chung","cross_cats":["math.OA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2016-11-27T05:09:08Z","title":"Quantitative $K$-theory for Banach algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.08790","kind":"arxiv","version":2},"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":"sha256:492e52327be1e6078b82be70dd9c6aaebacc7a79c93c9596d284dcdd358edd4e","target":"record","created_at":"2026-05-18T00:19:53Z","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":"e9da5232738b0a61714ba09753276e75f6fd2a606811791f24562b0a9992c2e5","cross_cats_sorted":["math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2016-11-27T05:09:08Z","title_canon_sha256":"41102c4491dde678cc0d62001e276a104e55f8350381ebd188657a99c5bef02d"},"schema_version":"1.0","source":{"id":"1611.08790","kind":"arxiv","version":2}},"canonical_sha256":"98d02ddab539e59aaf4b0e1d017b5dabddea6b0619c1ef709dc7c2d18691751f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"98d02ddab539e59aaf4b0e1d017b5dabddea6b0619c1ef709dc7c2d18691751f","first_computed_at":"2026-05-18T00:19:53.839738Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:19:53.839738Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HHbo7vHZRHnEzHU7yc0xMHbv10IrRTcCXMhpqCXja2ZYnqVWCV+BovbdToJh1yxLAJEwlFwmULKV9Pce4JEuBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:19:53.840446Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.08790","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:492e52327be1e6078b82be70dd9c6aaebacc7a79c93c9596d284dcdd358edd4e","sha256:e6edd0f2a74d28194ca791ad7caac7064742db9c0d663af19286fe8d1c9a5151"],"state_sha256":"aa6a386b725ea1269c6fba6adbd08991a24718ea3af59ef5be3b6f8a55564219"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Tk23vQxHlXo9ybW+lfhhg9EOWi+mHgMUnkQVT9wc1vQsOmPdZUuiVGgYbxCzbDmLnJou7qXtZkoR5kGxffM6Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T14:32:42.053321Z","bundle_sha256":"a0bc16c59a2b2fbe784bc922c4b513a96d863c1fadbc9c5c042a8afac56f4c89"}}