{ "schema": "exact.hp.public-facing-card-candidate.v1", "stable_card_id": "a3bcf63be1be2f724252fe0b1f11393935aafaea9c2f2fa70189d23c11bb0f65", "content": { "assumptions": "Let A ⊆ B(Hₐₜ) be the fixed unital C*-algebra obtained by adjoining the chosen shell-link unitaries to the atomic representation of the CAR algebra C, and then taking the norm-closed unital *-algebra they generate. Let K be any separable complex Hilbert space, and let σ: A → B(K) be any complex-linear multiplicative *-preserving map. It need not preserve the unit, and no faithfulness is assumed.", "conclusion": "σ is not both nonzero and irreducible. Equivalently, no such K and σ give a nonzero irreducible representation of A.", "content_completeness": "CARD_CONTENT_COMPLETE", "definition_explanation": null, "display_title": "No nonzero irreducible representation on a separable Hilbert space", "language_tag": "en", "lean_realization_notes": "The source predicate IsIrreducible includes nonzeroness, so the zero map is no counterexample to this statement. K ranges over an arbitrary independent universe v. This is not a claim that A is nonprimitive: its faithful atomic irreducible representation remains available. The comma in ∀ σ, ¬ σ.IsIrreducible separates the quantified variable from the proposition; it is not a substitute for an implication arrow.", "natural_language_statement": "The fixed CAR-based C*-algebra A has no nonzero irreducible representation on any separable complex Hilbert space.", "one_sentence_role": "Distinguishes separable representability from separable irreducible representability.", "proof_steps": [ { "step_id": "1", "text": "Convert the assumed nonzero irreducible comparison map to a unital representation." }, { "step_id": "2", "text": "Compare it and all other irreducible representations through the atomic model, obtaining the required singleton property." }, { "step_id": "3", "text": "Apply the separable singleton finite-dimensionality theorem and contradict the infinite-dimensionality of A established by the counterexample theorem for A." } ], "proof_summary": "If σ were nonzero and irreducible, it would be unital. The uniqueness of the atomic irreducible model would make it a separable singleton model, so the proved Rosenberg consequence would force A to be finite-dimensional, a contradiction." }, "card_ref": { "record_type": "LFH_DECLARATION_CARD", "revision": 2, "sha256": "ad091ef1bdb46599561192f9b2e32004e2aa24220ab5bf053514f7acb52b56d2", "uid": "lfh:lfh-declaration-card:sha256:a3bcf63be1be2f724252fe0b1f11393935aafaea9c2f2fa70189d23c11bb0f65" }, "exposition_ref": { "record_type": "LFH_CARD_EXPOSITION", "revision": 2, "sha256": "7cae6ae85cfa8d9830c5ec577b007152920d0fc6cce306be0afb5060d09ba7df", "uid": "lfh:lfh-card-exposition:sha256:4158dd23cb48e0a2980e0c7de01b999be64f1c9610e1a99ac5c189dc95dd5170" }, "source_binding": { "declaration_kind_from_source": "theorem", "qualified_name": "MathlibAnnex.CStarAlgebra.CAR.not_isIrreducible_of_separable", "source_commit": "437e6e46228bbb8e91211ebded349d7a30020e73", "source_lines": [ 24, 56 ], "source_path": "MathlibAnnex/Analysis/CStarAlgebra/CAR/SeparableIrreducible.lean", "source_repository": "r-tanaka-math/mathlib-annex", "source_url": "https://github.com/r-tanaka-math/mathlib-annex/blob/437e6e46228bbb8e91211ebded349d7a30020e73/MathlibAnnex/Analysis/CStarAlgebra/CAR/SeparableIrreducible.lean#L24-L56" }, "citations": [ { "citation_kind": "SOURCE_ATTRIBUTION", "target": { "kind": "EXACT_PROVIDER_LOCATOR", "module_name": "MathlibAnnex.Analysis.CStarAlgebra.CAR.AtomicCounterexample", "provider": "PROJECT", "provider_source_sha256": "5622f3f3cf8c469547e3e2584e0134f5204f202fa7a520217a5abcdb9a2a49e1", "qualified_name": "MathlibAnnex.CStarAlgebra.CAR.atomicCounterexampleEndpoint" } }, { "citation_kind": "SOURCE_ATTRIBUTION", "target": { "kind": "EXACT_PROVIDER_LOCATOR", "module_name": "MathlibAnnex.Analysis.CStarAlgebra.Representation.OrdinarySingleton", "provider": "PROJECT", "provider_source_sha256": "f9edc117f1607c33a7869f199cd685ce42e6baf1ce56c7510456d65aa78c5bdd", "qualified_name": "MathlibAnnex.Analysis.CStarAlgebra.Representation.finiteDimensional_algebra_of_singleton_amongNonUnital" } } ], "presentation": { "assumptions": "Let A ⊆ B(Hₐₜ) be the fixed unital C*-algebra obtained by adjoining the chosen shell-link unitaries to the atomic representation of the CAR algebra C, and then taking the norm-closed unital *-algebra they generate. Let K be any separable complex Hilbert space, and let σ: A → B(K) be any complex-linear multiplicative *-preserving map. It need not preserve the unit, and no faithfulness is assumed.", "conclusion": "σ is not both nonzero and irreducible. Equivalently, no such K and σ give a nonzero irreducible representation of A.", "content_completeness": "CARD_CONTENT_COMPLETE", "definition_explanation": null, "display_title": "No nonzero irreducible representation on a separable Hilbert space", "language_tag": "en", "lean_realization_notes": "The source predicate IsIrreducible includes nonzeroness, so the zero map is no counterexample to this statement. K ranges over an arbitrary independent universe v. This is not a claim that A is nonprimitive: its faithful atomic irreducible representation remains available. The comma in ∀ σ, ¬ σ.IsIrreducible separates the quantified variable from the proposition; it is not a substitute for an implication arrow.", "natural_language_statement": "The fixed CAR-based C*-algebra A has no nonzero irreducible representation on any separable complex Hilbert space.", "one_sentence_role": "Distinguishes separable representability from separable irreducible representability.", "proof_steps": [ { "step_id": "1", "text": "Convert the assumed nonzero irreducible comparison map to a unital representation." }, { "step_id": "2", "text": "Compare it and all other irreducible representations through the atomic model, obtaining the required singleton property." }, { "step_id": "3", "text": "Apply the separable singleton finite-dimensionality theorem and contradict the infinite-dimensionality of A established by the counterexample theorem for A." } ], "proof_summary": "If σ were nonzero and irreducible, it would be unital. The uniqueness of the atomic irreducible model would make it a separable singleton model, so the proved Rosenberg consequence would force A to be finite-dimensional, a contradiction.", "stable_card_id": "a3bcf63be1be2f724252fe0b1f11393935aafaea9c2f2fa70189d23c11bb0f65", "name": "MathlibAnnex.CStarAlgebra.CAR.not_isIrreducible_of_separable", "declaration_kind": "theorem", "source_path": "MathlibAnnex/Analysis/CStarAlgebra/CAR/SeparableIrreducible.lean", "source_locator": { "start_line": 24, "end_line": 56 }, "source_signature": { "text": "/-- The main target admits no nonzero irreducible ordinary representation on\na separable complete complex Hilbert space. The input representation is not\nassumed to preserve the unit. -/\ntheorem not_isIrreducible_of_separable\n {H : Type v} [NormedAddCommGroup H] [InnerProductSpace ℂ H]\n [CompleteSpace H] [TopologicalSpace.SeparableSpace H]\n (rho : NonUnitalRepresentation (A := AtomicCounterexampleAlgebra) (H := H)) :\n ¬ rho.IsIrreducible := by\n intro hrho\n let pi : Representation AtomicCounterexampleAlgebra H := rho.toUnital hrho\n have hpi : Representation.IsIrreducible pi :=\n NonUnitalRepresentation.isIrreducible_toUnital rho hrho\n have hambient_pi : atomicCounterexampleRepresentation.UnitaryEquivalent pi := by\n obtain ⟨U, hU⟩ := (atomicCounterexampleEndpoint.{v}).captures_nonunital H rho hrho\n refine ⟨U, ?_⟩\n intro a x\n simpa [atomicCounterexampleRepresentation, pi] using hU a x\n have hsingleton :\n Representation.IsSingletonIrreducibleModelAmongNonUnital.{0, v, 0} pi := by\n refine ⟨hpi, ?_⟩\n intro K _ _ _ sigma hsigma\n have hambient_sigma :\n atomicCounterexampleRepresentation.UnitaryEquivalent (sigma.toUnital hsigma) := by\n obtain ⟨U, hU⟩ := (atomicCounterexampleEndpoint.{0}).captures_nonunital K sigma hsigma\n refine ⟨U, ?_⟩\n intro a x\n change U (atomicCounterexampleRepresentation a x) = sigma a (U x)\n simpa [atomicCounterexampleRepresentation] using hU a x\n exact Representation.unitaryEquivalent_trans\n (Representation.unitaryEquivalent_symm hambient_pi) hambient_sigma\n exact (atomicCounterexampleEndpoint.{v}).not_finiteDimensional_target\n (Representation.finiteDimensional_algebra_of_singleton_amongNonUnital\n pi hsingleton)" }, "card_ref": { "record_type": "LFH_DECLARATION_CARD", "revision": 2, "sha256": "ad091ef1bdb46599561192f9b2e32004e2aa24220ab5bf053514f7acb52b56d2", "uid": "lfh:lfh-declaration-card:sha256:a3bcf63be1be2f724252fe0b1f11393935aafaea9c2f2fa70189d23c11bb0f65" }, "exposition_ref": { "record_type": "LFH_CARD_EXPOSITION", "revision": 2, "sha256": "7cae6ae85cfa8d9830c5ec577b007152920d0fc6cce306be0afb5060d09ba7df", "uid": "lfh:lfh-card-exposition:sha256:4158dd23cb48e0a2980e0c7de01b999be64f1c9610e1a99ac5c189dc95dd5170" }, "citations": [ { "label": "MathlibAnnex.CStarAlgebra.CAR.atomicCounterexampleEndpoint", "text": "Exact source attribution.", "website_href": "../fd0aa4fb4ef7616c41e13cea03169a5a1011d24ad2c287adbff4903787fd5470/index.html" }, { "label": "MathlibAnnex.Analysis.CStarAlgebra.Representation.finiteDimensional_algebra_of_singleton_amongNonUnital", "text": "Exact source attribution.", "website_href": "../../sources/naimark20/864b266d042477fabaee12ea/index.html#L96" } ] }, "website_metadata": { "publication_state": "PUBLIC_CURRENT", "publication_authorized": true, "first_publication_date": "2026-09-24", "this_revision_publication_date": "2026-09-24", "source_release": "v0.4.0", "source_exposition_correspondence": "APPROVED", "canonical_url": "https://exactmathematics.org/mathlibannex/cards/a3bcf63be1be2f724252fe0b1f11393935aafaea9c2f2fa70189d23c11bb0f65/", "catalog_current": true, "approval_scope": "APPROVED_SOURCE_EXPOSITION_CORRESPONDENCE", "publication_batch": "EM_NAIMARK_ARXIV_CARD20_PUBLICATION_R1" } }