{
  "accepted_type_fingerprint_sha256": "6f51d3541879258dec9a02a0c0a1c7992002b0287ba3a17257eb5449b5850f8f",
  "approved_card_model_sha256": "26097183889177152c27d06f6f69fb9fcf6c1bf801d99efb1c9fb4918793ae74",
  "assumptions": "Let A be a nonzero complex C*-algebra, not assumed unital, H a separable complex Hilbert space, and π a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. No simplicity assumption is made.",
  "card_ref": {
    "record_type": "LFH_DECLARATION_CARD",
    "revision": 1,
    "sha256": "9c84c41fbd329d019a48250459bbaa298670bcb68317e4f2c154f1c9e5144ac3",
    "uid": "lfh:lfh-declaration-card:sha256:be93429f16421fb0cf8eb8bb91861473c1fd7b52af3892ee1ff2d10e1fb20c85"
  },
  "citations": [
    {
      "label": "MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation.exists_preimage_rankOne_of_singleton",
      "target_card_ref": {
        "record_type": "LFH_DECLARATION_CARD",
        "revision": 2,
        "sha256": "a1c2ac8a3243c4697c1d5133870dfbda664c5a4e32cfba5f403fe5d6f387df5a",
        "uid": "lfh:lfh-declaration-card:sha256:155676cd74606a102aa274be40c9fb294971cfef6df06897084c9fc89322f2cb"
      },
      "text": "Exact formal dependency; inspect the linked Card and exact source."
    },
    {
      "label": "MathlibAnnex.Analysis.CStarAlgebra.exists_character_annihilating_closedIdeal_of_not_mem",
      "target_card_ref": {
        "record_type": "LFH_DECLARATION_CARD",
        "revision": 2,
        "sha256": "6478c2932ffa998eda8ac724e6df08a7c18971e943cf48ac511aa19e7793261a",
        "uid": "lfh:lfh-declaration-card:sha256:1e09b496fe0e06eeb046e141957245bfb51a6b1b7b8c28cf44ffdefc9574de2f"
      },
      "text": "Exact formal dependency; inspect the linked Card and exact source."
    },
    {
      "label": "MathlibAnnex.CStarAlgebra.compactPreimageIdeal",
      "target_card_ref": {
        "record_type": "LFH_DECLARATION_CARD",
        "revision": 2,
        "sha256": "ee21384769079014e7f0b83108c683e777408b5e63149385e2931aa44db2b8b1",
        "uid": "lfh:lfh-declaration-card:sha256:9355a7df6defbdd54a898d97fa5d5ff61ac59614adb1ceee96df088d9c1a66b6"
      },
      "text": "Exact formal dependency; inspect the linked Card and exact source."
    }
  ],
  "conclusion": "Every operator π(a), a∈A, is compact.",
  "declaration_kind": "theorem",
  "definition_equation": "∀ a : A, IsCompactOperator (π a).",
  "definition_explanation": "∀ a : A, IsCompactOperator (π a).",
  "display_title": "Compact image of a representative of the unique irreducible-representation class",
  "exact_elaborated_lean_type": "∀ {A : Type u} [inst : NonUnitalCStarAlgebra A] [inst_1 : PartialOrder A] [StarOrderedRing A] {H : Type v} [inst_3 : NormedAddCommGroup H] [inst_4 : InnerProductSpace ℂ H] [inst_5 : CompleteSpace H] [Nontrivial A] [TopologicalSpace.SeparableSpace H] (pi : MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation A H), pi.IsSingletonIrreducibleModel → ∀ (a : A), IsCompactOperator ⇑(pi a)",
  "exposition_ref": {
    "record_type": "LFH_CARD_EXPOSITION",
    "revision": 6,
    "sha256": "8c3c5f77b4171d3fa81919e7f7f2167d426edcc10c67c36e31a82665eadfe93c",
    "uid": "lfh:lfh-card-exposition:sha256:b40ecd96453faf3bf4795905f06c397865b27b8a1dd854d7459fee7eb7410e90"
  },
  "historical_card_ref": {
    "record_type": "LFH_DECLARATION_CARD",
    "revision": 1,
    "sha256": "c1a0d4bdcf857802f724dc08c62ac72a2889783ae34ce8fec2291ce05f9a86f6",
    "uid": "lfh:lfh-declaration-card:sha256:94b7f2220243cb53c84a521c24f6eb1f35bc8222d4162b9ea134b4c10e9ea7b4"
  },
  "historical_exposition_ref": {
    "record_type": "LFH_CARD_EXPOSITION",
    "revision": 6,
    "sha256": "8c3c5f77b4171d3fa81919e7f7f2167d426edcc10c67c36e31a82665eadfe93c",
    "uid": "lfh:lfh-card-exposition:sha256:b40ecd96453faf3bf4795905f06c397865b27b8a1dd854d7459fee7eb7410e90"
  },
  "historical_owner_decision": "APPROVED_SOURCE_EXPOSITION_CORRESPONDENCE_WITHOUT_CORRECTION",
  "lineage": {
    "classification": "DISTINCT_UID_REVISION_ONE_RENAMED_TO",
    "predecessor_card_ref": {
      "record_type": "LFH_DECLARATION_CARD",
      "revision": 1,
      "sha256": "c1a0d4bdcf857802f724dc08c62ac72a2889783ae34ce8fec2291ce05f9a86f6",
      "uid": "lfh:lfh-declaration-card:sha256:94b7f2220243cb53c84a521c24f6eb1f35bc8222d4162b9ea134b4c10e9ea7b4"
    },
    "renamed_to_relation_ref": {
      "record_type": "LFH_DECLARATION_RELATION",
      "revision": 1,
      "sha256": "c0bcad8bba782a98b79c3c4aa0d946f30e88e62e4b99d82f6263f09a89c35e2c",
      "uid": "lfh:lfh-declaration-relation:sha256:d7f8237da04ff9414301e3feff015fb016a23283ce8ff66bc99438ef7f427207"
    }
  },
  "name": "MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation.isCompactOperator_map_of_singleton",
  "natural_language_statement": "Let A be a nonzero complex C*-algebra, not assumed unital, H a separable complex Hilbert space, and π a representation of A on H that is a representative of the unique unitary-equivalence class of nonzero irreducible representations of A. No simplicity assumption is made. Every operator π(a), a∈A, is compact.",
  "one_sentence_role": "Shows that every operator in the image of a separably acting representative of the unique irreducible-representation class is compact.",
  "owner_review_item_sha256": "5cf4dcb1dd019784ff2051a9b8e4be5a9800e79645bb38a610287d341fe55642",
  "proof_steps": [
    {
      "step_id": "formal_type",
      "text": "Exact Lean statement: ∀ {A : Type u} [inst : NonUnitalCStarAlgebra A] [inst_1 : PartialOrder A] [StarOrderedRing A] {H : Type v} [inst_3 : NormedAddCommGroup H] [inst_4 : InnerProductSpace ℂ H] [inst_5 : CompleteSpace H] [Nontrivial A] [TopologicalSpace.SeparableSpace H] (pi : MathlibAnnex.Analysis.CStarAlgebra.NonUnitalCStarRepresentation A H), pi.IsSingletonIrreducibleModel → ∀ (a : A), IsCompactOperator ⇑(pi a)"
    },
    {
      "step_id": "source_equation",
      "text": "∀ a : A, IsCompactOperator (π a)."
    },
    {
      "step_id": "source_route",
      "text": "Assuming a noncompact image, the source takes a noncompact self-adjoint part and embeds it in a maximal abelian unitization subalgebra D. The closed compact-preimage ideal in D does not contain that element; character separation produces χ. The eigenvector associated with a character distinct from the scalar character and a compact rank-one projection contradict χ's annihilation of the ideal. This follows the source's character route, not a norm-separable pure-state plan."
    }
  ],
  "proof_summary": "Assuming a noncompact image, the source takes a noncompact self-adjoint part and embeds it in a maximal abelian unitization subalgebra D. The closed compact-preimage ideal in D does not contain that element; character separation produces χ. The eigenvector associated with a character distinct from the scalar character and a compact rank-one projection contradict χ's annihilation of the ideal. This follows the source's character route, not a norm-separable pure-state plan.",
  "source_excerpt": {
    "bytes": 5004,
    "sha256": "27897ba5b0cef51899d65d7bdbd318b92c6fff2b8c6d63841d2ec3033bd866c5"
  },
  "source_locator": {
    "end_column": 43,
    "end_line": 182,
    "source_path": "MathlibAnnex/Analysis/CStarAlgebra/NonUnital/CompactExclusion.lean",
    "source_sha256": "faa2c79a30832771558546b265481337a38256f44920a71b29da9b8d8db659a2",
    "start_column": 1,
    "start_line": 68
  },
  "source_path": "MathlibAnnex/Analysis/CStarAlgebra/NonUnital/CompactExclusion.lean",
  "source_realization_ref": {
    "record_type": "LFH_SOURCE_BUILD_BINDING",
    "revision": 1,
    "sha256": "f805d9fc741af5f4946cfea40e453fa231b982558fc4868e7dfb2de11e435d8f",
    "uid": "lfh:lfh-source-build-binding:sha256:9fbc6221605e13d19d23fa06a27fa2341871bbc8e2716d3b5d3a56486e6cfea1"
  },
  "source_sha256": "faa2c79a30832771558546b265481337a38256f44920a71b29da9b8d8db659a2",
  "source_signature": {
    "end_byte": 3186,
    "extraction_profile": "PARSED_COMMAND_TOP_LEVEL_DECLARATION_HEADER_V1",
    "sha256": "d414351435e9a7b6aeaa8fe899fd1d514447a6090d376a69710eccd745f73e7f",
    "source_command_range": {
      "end_column": 43,
      "end_line": 182,
      "source_path": "MathlibAnnex/Analysis/CStarAlgebra/NonUnital/CompactExclusion.lean",
      "source_sha256": "faa2c79a30832771558546b265481337a38256f44920a71b29da9b8d8db659a2",
      "start_column": 1,
      "start_line": 68
    },
    "source_path": "MathlibAnnex/Analysis/CStarAlgebra/NonUnital/CompactExclusion.lean",
    "source_sha256": "faa2c79a30832771558546b265481337a38256f44920a71b29da9b8d8db659a2",
    "start_byte": 2946,
    "text": "theorem isCompactOperator_map_of_singleton [Nontrivial A]\n    [TopologicalSpace.SeparableSpace H]\n    (pi : NonUnitalCStarRepresentation A H)\n    (hsingle : IsSingletonIrreducibleModel.{u, v, u} pi) :\n    ∀ a : A, IsCompactOperator (pi a)"
  },
  "source_url": "https://github.com/r-tanaka-math/mathlib-annex/blob/437e6e46228bbb8e91211ebded349d7a30020e73/MathlibAnnex/Analysis/CStarAlgebra/NonUnital/CompactExclusion.lean#L68-L182",
  "stable_card_id": "be93429f16421fb0cf8eb8bb91861473c1fd7b52af3892ee1ff2d10e1fb20c85",
  "successor_owner_approval": "APPROVED_REUSED_SOURCE_EXPOSITION_CORRESPONDENCE",
  "website_metadata": {
    "source_release": "v0.4.0",
    "catalog_scope": "WHOLE_LIBRARY",
    "publication_authorized": true,
    "catalog_current": true,
    "first_publication_date": "2026-09-21",
    "canonical_model_sha256": "7977ace875db05782fdc2f7034c811a9253728a7662bdf5bb9506a70d6cf6696",
    "publication_state": "PUBLIC",
    "this_revision_publication_date": "2026-09-21",
    "canonical_url": "https://exactmathematics.org/mathlibannex/cards/be93429f16421fb0cf8eb8bb91861473c1fd7b52af3892ee1ff2d10e1fb20c85/",
    "publication_batch": "EM_FULL_HP_PUBLICATION_R1",
    "approval_scope": "APPROVED_REUSED_SOURCE_EXPOSITION_CORRESPONDENCE"
  }
}