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Dales, Niels Jakob Laustsen, Piotr Koszmider, Tomasz Kania, Tomasz Kochanek","submitted_at":"2012-08-23T14:13:06Z","abstract_excerpt":"We address the following two questions regarding the maximal left ideals of the Banach algebra $\\mathscr{B}(E)$ of bounded operators acting on an infinite-dimensional Banach pace $E$:\n  (Q1) Does $\\mathscr{B}(E)$ always contain a maximal left ideal which is not finitely generated? (Q2) Is every finitely-generated, maximal left ideal of $\\mathscr{B}(E)$ necessarily of the form \\{T\\in\\mathscr{B}(E): Tx = 0\\} (*) for some non-zero $x\\in E$?\n  Since the two-sided ideal $\\mathscr{F}(E)$ of finite-rank operators is not contained in any of the maximal left ideals given by (*), a positive answer to th"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1208.4762","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-08-23T14:13:06Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"a5e7faea3b21dab9d4b271e39d2681d0c0737827ac5e41b41c9f25ebcad8e6a3","abstract_canon_sha256":"f72d11b025af60a279221ba5fbdb05f10ec17936967ce14a39ea8cd5416e7e3f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:32:54.616403Z","signature_b64":"rWwBaNHjPBa/KKC5r9CFMmMHc9MrHFdZfTKrboa+kdha+2MMWAvj7zvX5YgaWzYTPWnqO/iWCUM00HKTylG+Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c367dd579d2d40ac6fb82af9b0565f05dfd03e8ae40bf64c4e9b25aa239f51b4","last_reissued_at":"2026-05-18T02:32:54.616044Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:32:54.616044Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Maximal left ideals of the Banach algebra of bounded operators on a Banach space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"H. G. Dales, Niels Jakob Laustsen, Piotr Koszmider, Tomasz Kania, Tomasz Kochanek","submitted_at":"2012-08-23T14:13:06Z","abstract_excerpt":"We address the following two questions regarding the maximal left ideals of the Banach algebra $\\mathscr{B}(E)$ of bounded operators acting on an infinite-dimensional Banach pace $E$:\n  (Q1) Does $\\mathscr{B}(E)$ always contain a maximal left ideal which is not finitely generated? (Q2) Is every finitely-generated, maximal left ideal of $\\mathscr{B}(E)$ necessarily of the form \\{T\\in\\mathscr{B}(E): Tx = 0\\} (*) for some non-zero $x\\in E$?\n  Since the two-sided ideal $\\mathscr{F}(E)$ of finite-rank operators is not contained in any of the maximal left ideals given by (*), a positive answer to th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.4762","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1208.4762","created_at":"2026-05-18T02:32:54.616093+00:00"},{"alias_kind":"arxiv_version","alias_value":"1208.4762v3","created_at":"2026-05-18T02:32:54.616093+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.4762","created_at":"2026-05-18T02:32:54.616093+00:00"},{"alias_kind":"pith_short_12","alias_value":"YNT52V45FVAK","created_at":"2026-05-18T12:27:27.928770+00:00"},{"alias_kind":"pith_short_16","alias_value":"YNT52V45FVAKY35Y","created_at":"2026-05-18T12:27:27.928770+00:00"},{"alias_kind":"pith_short_8","alias_value":"YNT52V45","created_at":"2026-05-18T12:27:27.928770+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/YNT52V45FVAKY35YFL43AVS7AX","json":"https://pith.science/pith/YNT52V45FVAKY35YFL43AVS7AX.json","graph_json":"https://pith.science/api/pith-number/YNT52V45FVAKY35YFL43AVS7AX/graph.json","events_json":"https://pith.science/api/pith-number/YNT52V45FVAKY35YFL43AVS7AX/events.json","paper":"https://pith.science/paper/YNT52V45"},"agent_actions":{"view_html":"https://pith.science/pith/YNT52V45FVAKY35YFL43AVS7AX","download_json":"https://pith.science/pith/YNT52V45FVAKY35YFL43AVS7AX.json","view_paper":"https://pith.science/paper/YNT52V45","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1208.4762&json=true","fetch_graph":"https://pith.science/api/pith-number/YNT52V45FVAKY35YFL43AVS7AX/graph.json","fetch_events":"https://pith.science/api/pith-number/YNT52V45FVAKY35YFL43AVS7AX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YNT52V45FVAKY35YFL43AVS7AX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YNT52V45FVAKY35YFL43AVS7AX/action/storage_attestation","attest_author":"https://pith.science/pith/YNT52V45FVAKY35YFL43AVS7AX/action/author_attestation","sign_citation":"https://pith.science/pith/YNT52V45FVAKY35YFL43AVS7AX/action/citation_signature","submit_replication":"https://pith.science/pith/YNT52V45FVAKY35YFL43AVS7AX/action/replication_record"}},"created_at":"2026-05-18T02:32:54.616093+00:00","updated_at":"2026-05-18T02:32:54.616093+00:00"}