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Let \\[ \\zeta\\otimes B : \\qquad P \\times_G B \\longrightarrow S^k \\] denote the associated bundle and let \\[ A_{\\zeta\\otimes B} = \\Gamma (S^k, P \\times_G B) \\] denote the associated Banach algebra of sections. Then $\\pi_*\\GL A_{\\zeta \\otimes B} $ is determined by a mostly degenerate spectral sequence and by a Wang differential \\[ d_k : \\pi_*(\\GL B) \\longrightarrow \\pi_{*+k-1} (\\GL B) .\\] We show that if $B$ is a $C^*$-algebra then the differential is given explic"},"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":"1209.4131","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-09-19T01:40:59Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"6a1abd801844f4433349ae8b1bcb8aa39f5b4c4fa014a11d3f4e0ad0b3f50f46","abstract_canon_sha256":"1f8b92c21b86dfcd9dfa4beef3ff27fa4421d69a335bea3846f8dd46d9c192d7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:22.141467Z","signature_b64":"a0zm5HxGUg+VfEGfrPw9TAcz05tKzIoiHAup4NbR4SnqANOM3hNPTiEW5ecju58usytP3rGadrG56FGuEwzzAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e17e304d66a86b268df879b119804ea929097084ba4641c73b814ecbca36ec82","last_reissued_at":"2026-05-18T02:58:22.140956Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:22.140956Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Banach algebras, Samelson products, and the Wang Differential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.OA","authors_text":"Claude L. 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Then $\\pi_*\\GL A_{\\zeta \\otimes B} $ is determined by a mostly degenerate spectral sequence and by a Wang differential \\[ d_k : \\pi_*(\\GL B) \\longrightarrow \\pi_{*+k-1} (\\GL B) .\\] We show that if $B$ is a $C^*$-algebra then the differential is given explic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.4131","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":"1209.4131","created_at":"2026-05-18T02:58:22.141038+00:00"},{"alias_kind":"arxiv_version","alias_value":"1209.4131v3","created_at":"2026-05-18T02:58:22.141038+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.4131","created_at":"2026-05-18T02:58:22.141038+00:00"},{"alias_kind":"pith_short_12","alias_value":"4F7DATLGVBVS","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_16","alias_value":"4F7DATLGVBVSNDPY","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_8","alias_value":"4F7DATLG","created_at":"2026-05-18T12:26:53.410803+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/4F7DATLGVBVSNDPYPGYRTACOVE","json":"https://pith.science/pith/4F7DATLGVBVSNDPYPGYRTACOVE.json","graph_json":"https://pith.science/api/pith-number/4F7DATLGVBVSNDPYPGYRTACOVE/graph.json","events_json":"https://pith.science/api/pith-number/4F7DATLGVBVSNDPYPGYRTACOVE/events.json","paper":"https://pith.science/paper/4F7DATLG"},"agent_actions":{"view_html":"https://pith.science/pith/4F7DATLGVBVSNDPYPGYRTACOVE","download_json":"https://pith.science/pith/4F7DATLGVBVSNDPYPGYRTACOVE.json","view_paper":"https://pith.science/paper/4F7DATLG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1209.4131&json=true","fetch_graph":"https://pith.science/api/pith-number/4F7DATLGVBVSNDPYPGYRTACOVE/graph.json","fetch_events":"https://pith.science/api/pith-number/4F7DATLGVBVSNDPYPGYRTACOVE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4F7DATLGVBVSNDPYPGYRTACOVE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4F7DATLGVBVSNDPYPGYRTACOVE/action/storage_attestation","attest_author":"https://pith.science/pith/4F7DATLGVBVSNDPYPGYRTACOVE/action/author_attestation","sign_citation":"https://pith.science/pith/4F7DATLGVBVSNDPYPGYRTACOVE/action/citation_signature","submit_replication":"https://pith.science/pith/4F7DATLGVBVSNDPYPGYRTACOVE/action/replication_record"}},"created_at":"2026-05-18T02:58:22.141038+00:00","updated_at":"2026-05-18T02:58:22.141038+00:00"}