OAI/Analysis/FactorGeneration/MatrixUnits.lean

GitHub source file

56 declarations, 0 contain sorry, 0 declared as axioms

Actual finite matrix algebras from matrix-unit laws. No factor or trace hypothesis is needed for the algebraic construction.

  1. structure OAI.Generator.MatrixUnitSystem line 18
  2. def OAI.Generator.MatrixUnitSystem.linearMap line 30
  3. theorem OAI.Generator.MatrixUnitSystem.linearMap_apply line 35
  4. theorem OAI.Generator.MatrixUnitSystem.linearMap_single line 38
  5. theorem OAI.Generator.MatrixUnitSystem.linearMap_one line 42
  6. theorem OAI.Generator.MatrixUnitSystem.linearMap_mul line 45
  7. theorem OAI.Generator.MatrixUnitSystem.linearMap_star line 61
  8. def OAI.Generator.MatrixUnitSystem.hom line 67
  9. theorem OAI.Generator.MatrixUnitSystem.hom_apply line 78
  10. theorem OAI.Generator.MatrixUnitSystem.hom_single line 81
  11. theorem OAI.Generator.MatrixUnitSystem.sandwich line 84
  12. theorem OAI.Generator.MatrixUnitSystem.hom_injective line 90
  13. def OAI.Generator.MatrixUnitSystem.algebra line 97
  14. theorem OAI.Generator.MatrixUnitSystem.mem_algebra line 99
  15. theorem OAI.Generator.MatrixUnitSystem.entry_mem line 102
  16. def OAI.Generator.MatrixUnitSystem.equiv line 105
  17. instance OAI.Generator.MatrixUnitSystem.algebra_finiteDimensional line 108
  18. theorem OAI.Generator.MatrixUnitSystem.algebra_isClosed line 113
  19. theorem OAI.Generator.MatrixUnitSystem.diagonal_ne_zero line 127
  20. theorem OAI.Generator.MatrixUnitSystem.injective line 139
  21. theorem OAI.Generator.MatrixUnitSystem.algebra_le_iff line 142
  22. theorem OAI.Generator.MatrixUnitSystem.commute_algebra_iff line 150
  23. def OAI.Generator.MatrixUnitSystem.product line 163
  24. theorem OAI.Generator.MatrixUnitSystem.product_entry line 182
  25. theorem OAI.Generator.MatrixUnitSystem.product_sum_left line 186
  26. theorem OAI.Generator.MatrixUnitSystem.product_sum_right line 191
  27. theorem OAI.Generator.MatrixUnitSystem.algebra_le_product_left line 195
  28. theorem OAI.Generator.MatrixUnitSystem.algebra_le_product_right line 201
  29. theorem OAI.Generator.MatrixUnitSystem.product_algebra_eq_sup line 207
  30. theorem OAI.Generator.MatrixUnitSystem.diagonal_sum_retained line 216
  31. def OAI.Generator.MatrixUnitSystem.commutant line 230
  32. theorem OAI.Generator.MatrixUnitSystem.mem_commutant line 242
  33. def OAI.Generator.MatrixUnitSystem.amplify line 244
  34. theorem OAI.Generator.MatrixUnitSystem.entry_mul_amplify line 247
  35. theorem OAI.Generator.MatrixUnitSystem.amplify_mul_entry line 253
  36. theorem OAI.Generator.MatrixUnitSystem.amplify_mem line 258
  37. theorem OAI.Generator.MatrixUnitSystem.amplify_star line 264
  38. theorem OAI.Generator.MatrixUnitSystem.amplify_zero line 268
  39. theorem OAI.Generator.MatrixUnitSystem.amplify_corner line 271
  40. theorem OAI.Generator.MatrixUnitSystem.amplify_mul line 277
  41. theorem OAI.Generator.MatrixUnitSystem.amplify_compression line 286
  42. theorem OAI.Generator.MatrixUnitSystem.reconstruct line 297
  43. theorem OAI.Generator.MatrixUnitSystem.commutant_scalar_center line 309

    Source documentation

    The commutant of a full unital matrix algebra in a scalar-center algebra has scalar center. The finite matrix decomposition is constructed above.

    theorem commutant_scalar_center (o : ι)
    (hfactor : ∀ c : A, (∀ a : A,a*c=c*a) → ∃ z : ℂ,c=z • (1 : A))
    (c : E.commutant) (hc : ∀ a : E.commutant,a*c=c*a) :
    ∃ z : ℂ,c=z • (1 : E.commutant) := by
…
  44. theorem OAI.Generator.MatrixUnitSystem.diagonal_projection line 332
  45. theorem OAI.Generator.MatrixUnitSystem.amplify_projection line 336
  46. theorem OAI.Generator.MatrixUnitSystem.commutant_diffuse line 344
  47. def OAI.Generator.MatrixUnitSystem.map line 403
  48. theorem OAI.Generator.MatrixUnitSystem.map_entry line 411
  49. theorem OAI.Generator.MatrixUnitSystem.diagonal_sum_retained_right line 413
  50. def OAI.Generator.MatrixUnitSystem.coefficientEval line 429
  51. theorem OAI.Generator.MatrixUnitSystem.coefficientEval_mul line 432
  52. theorem OAI.Generator.MatrixUnitSystem.coefficientEval_star line 453
  53. theorem OAI.Generator.MatrixUnitSystem.coefficientEval_one line 464
  54. def OAI.Generator.MatrixUnitSystem.coefficientHom line 467
  55. theorem OAI.Generator.MatrixUnitSystem.coefficientHom_single line 482
  56. theorem OAI.Generator.MatrixUnitSystem.coefficientHom_surjective line 487