{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:ZX5BL54VUILIG2XDFQOW5WAAWL","short_pith_number":"pith:ZX5BL54V","canonical_record":{"source":{"id":"1212.0679","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-12-04T11:16:41Z","cross_cats_sorted":[],"title_canon_sha256":"1edaa6171615667c2cbadebab06ddb23ccd377a823e9baec93e107c11b0252d9","abstract_canon_sha256":"880b72d9a31ef6e4fd76bdd04399b741563dae1990df1ec72042877ec1ee16ad"},"schema_version":"1.0"},"canonical_sha256":"cdfa15f795a216836ae32c1d6ed800b2e80e47b5f1b905e1b2e2245e92497d7d","source":{"kind":"arxiv","id":"1212.0679","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.0679","created_at":"2026-05-18T03:39:15Z"},{"alias_kind":"arxiv_version","alias_value":"1212.0679v1","created_at":"2026-05-18T03:39:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.0679","created_at":"2026-05-18T03:39:15Z"},{"alias_kind":"pith_short_12","alias_value":"ZX5BL54VUILI","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"ZX5BL54VUILIG2XD","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"ZX5BL54V","created_at":"2026-05-18T12:27:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:ZX5BL54VUILIG2XDFQOW5WAAWL","target":"record","payload":{"canonical_record":{"source":{"id":"1212.0679","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-12-04T11:16:41Z","cross_cats_sorted":[],"title_canon_sha256":"1edaa6171615667c2cbadebab06ddb23ccd377a823e9baec93e107c11b0252d9","abstract_canon_sha256":"880b72d9a31ef6e4fd76bdd04399b741563dae1990df1ec72042877ec1ee16ad"},"schema_version":"1.0"},"canonical_sha256":"cdfa15f795a216836ae32c1d6ed800b2e80e47b5f1b905e1b2e2245e92497d7d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:39:15.502184Z","signature_b64":"MOTwszDtNKIeyKFhqifwfxCmMELSaxum4nw2C3q6/NYMX56qfKvBHK/GVwZ8SB9+NXMztGfjFhdUf+e3LoFwAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cdfa15f795a216836ae32c1d6ed800b2e80e47b5f1b905e1b2e2245e92497d7d","last_reissued_at":"2026-05-18T03:39:15.501609Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:39:15.501609Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1212.0679","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:39:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/CoiNsWOGLKa+MFlax+QfQ1NG3jcG8Q0sjQq7FBLExhT3KhumUnDeigThV+mzADXULD2uyZUg3zLcq4gXhuZDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T23:13:24.302822Z"},"content_sha256":"40985fbd25bac1e48cf5a9dc6c48de5854e04592f6a0c478b8862f5e02265f7b","schema_version":"1.0","event_id":"sha256:40985fbd25bac1e48cf5a9dc6c48de5854e04592f6a0c478b8862f5e02265f7b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:ZX5BL54VUILIG2XDFQOW5WAAWL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Homological Properties of the Algebra of Compact Operators on a Banach Space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"George A. Willis","submitted_at":"2012-12-04T11:16:41Z","abstract_excerpt":"The conditions on a Banach space, $E$, under which the algebra, $\\mathcal{K}(E)$, of compact operators on $E$ is right flat or homologically unital are investigated. These homological properties are related to factorization in the algebra and it is shown that, for $\\mathcal{K}(E)$, they are closely associated with the approximation property for $E$. The class of spaces, $E$, such that $\\mathcal{K}(E)$ is known to be right flat and homologically unital is extended to include spaces which do not have the bounded compact approximation property."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.0679","kind":"arxiv","version":1},"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":"2026-05-18T03:39:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5uHhJYYvmTWwtn1bczl78YVp9WUc8nOW+ZQCDZDR2N+lWeiyPdI+cMW3Nu7ova7yzdM16m33stXTvUY/NQETAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T23:13:24.303199Z"},"content_sha256":"db78eca3d534c38cfb2c4d897b36e1abb4b9eaba743b38efd05dc0ee604ed87b","schema_version":"1.0","event_id":"sha256:db78eca3d534c38cfb2c4d897b36e1abb4b9eaba743b38efd05dc0ee604ed87b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZX5BL54VUILIG2XDFQOW5WAAWL/bundle.json","state_url":"https://pith.science/pith/ZX5BL54VUILIG2XDFQOW5WAAWL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZX5BL54VUILIG2XDFQOW5WAAWL/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":"2026-06-11T23:13:24Z","links":{"resolver":"https://pith.science/pith/ZX5BL54VUILIG2XDFQOW5WAAWL","bundle":"https://pith.science/pith/ZX5BL54VUILIG2XDFQOW5WAAWL/bundle.json","state":"https://pith.science/pith/ZX5BL54VUILIG2XDFQOW5WAAWL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZX5BL54VUILIG2XDFQOW5WAAWL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:ZX5BL54VUILIG2XDFQOW5WAAWL","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":"880b72d9a31ef6e4fd76bdd04399b741563dae1990df1ec72042877ec1ee16ad","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-12-04T11:16:41Z","title_canon_sha256":"1edaa6171615667c2cbadebab06ddb23ccd377a823e9baec93e107c11b0252d9"},"schema_version":"1.0","source":{"id":"1212.0679","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.0679","created_at":"2026-05-18T03:39:15Z"},{"alias_kind":"arxiv_version","alias_value":"1212.0679v1","created_at":"2026-05-18T03:39:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.0679","created_at":"2026-05-18T03:39:15Z"},{"alias_kind":"pith_short_12","alias_value":"ZX5BL54VUILI","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"ZX5BL54VUILIG2XD","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"ZX5BL54V","created_at":"2026-05-18T12:27:30Z"}],"graph_snapshots":[{"event_id":"sha256:db78eca3d534c38cfb2c4d897b36e1abb4b9eaba743b38efd05dc0ee604ed87b","target":"graph","created_at":"2026-05-18T03:39:15Z","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":"The conditions on a Banach space, $E$, under which the algebra, $\\mathcal{K}(E)$, of compact operators on $E$ is right flat or homologically unital are investigated. These homological properties are related to factorization in the algebra and it is shown that, for $\\mathcal{K}(E)$, they are closely associated with the approximation property for $E$. The class of spaces, $E$, such that $\\mathcal{K}(E)$ is known to be right flat and homologically unital is extended to include spaces which do not have the bounded compact approximation property.","authors_text":"George A. Willis","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-12-04T11:16:41Z","title":"Homological Properties of the Algebra of Compact Operators on a Banach Space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.0679","kind":"arxiv","version":1},"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":"sha256:40985fbd25bac1e48cf5a9dc6c48de5854e04592f6a0c478b8862f5e02265f7b","target":"record","created_at":"2026-05-18T03:39:15Z","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":"880b72d9a31ef6e4fd76bdd04399b741563dae1990df1ec72042877ec1ee16ad","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-12-04T11:16:41Z","title_canon_sha256":"1edaa6171615667c2cbadebab06ddb23ccd377a823e9baec93e107c11b0252d9"},"schema_version":"1.0","source":{"id":"1212.0679","kind":"arxiv","version":1}},"canonical_sha256":"cdfa15f795a216836ae32c1d6ed800b2e80e47b5f1b905e1b2e2245e92497d7d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cdfa15f795a216836ae32c1d6ed800b2e80e47b5f1b905e1b2e2245e92497d7d","first_computed_at":"2026-05-18T03:39:15.501609Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:39:15.501609Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MOTwszDtNKIeyKFhqifwfxCmMELSaxum4nw2C3q6/NYMX56qfKvBHK/GVwZ8SB9+NXMztGfjFhdUf+e3LoFwAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:39:15.502184Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.0679","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:40985fbd25bac1e48cf5a9dc6c48de5854e04592f6a0c478b8862f5e02265f7b","sha256:db78eca3d534c38cfb2c4d897b36e1abb4b9eaba743b38efd05dc0ee604ed87b"],"state_sha256":"5cbada346cb089f8700c94f1a3067ee03b23dbf58e787b6d411de6b541b4cbd8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c+UN+xcouewzpgZ4j9Cbej8FJAD62JsS5vTU2x9EkM5uHbtM2fr4emlN2/idASH/H2i1riEVBAHm/Pb52ocwCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T23:13:24.306788Z","bundle_sha256":"9877b7c337e96ad49f8832efd9da7b4714e3d594c5f9646a5ff93cdff0ad9648"}}