Subfactors, index and finite-depth classification

A research-level course on the size and structure of factor inclusions, from the basic construction and finite bases to the standard invariant and finite-depth classification.

These lessons are draft chapters. The author has checked their written arguments relative to the stated course prerequisites; the full course is still being developed.

  1. Finite traces and Jones projections
  2. A projection that remembers an inclusion
  3. Finite algebras and normal traces
  4. Measuring an inclusion through modules and corners
  5. Hyperfinite corners and diagonal indices
  6. Finite bases, bounded vectors and a positive-operator inequality
  7. Going up and down the Jones tower
  8. Commuting projections need not be maximal abelian
  9. Fusion as a concrete operator algebra
  10. Positivity restricts the index
  11. Matrix inclusions and the Markov trace
  12. Paths, local projections and a faithful trace
  13. The trace and the tail of a path
  14. Removing a projection produces a subfactor
  15. Reflection, commuting squares and finite depth
  16. Why the discrete projection algebra is unique
  17. Reflected traces and a uniform bound along a tunnel
  18. Detecting a generating tunnel
  19. Matrix corners approximate a tunnel
  20. A generating tunnel and the classification theorem
  21. A finite symmetry group gives two kinds of index
  22. The principal graph records fusion multiplicities
  23. Graphs below norm two and a corner obstruction
  24. The dual weight and the Jones projection
  25. One expectation, one index in every representation
  26. A path graph determines the whole invariant
  27. Index, composition and localized observables
  28. A generating tail realizes the path invariant
  29. A cyclic symmetry realizes the three-armed graph
  30. Charges modulo three determine the whole invariant
  31. An odd fork contradicts integral fusion multiplicities
  32. A finite-index tunnel is enough
  33. A finite square produces an inclusion
  34. A finite angle determines the relative commutant
  35. Two unitary matrices build a path grid
  36. Flat paths recover the principal graph
  37. Folding a path into a fork
  38. A braid phase decides which forks are flat
  39. A branching matrix determines the connection
  40. An inclusion determines its connection
  41. Exact certificates make the exceptional graphs flat
  42. An ordered trace distinguishes the opposite exceptional inclusions
  43. When finite reflection extends to a tracial limit
  44. A group inclusion distinguishes the original pair from its dual
  45. Polynomials locate the finite projection gaps
  46. A full corner recognizes the standard basic construction
  47. Extending a finite projection algebra through the cutoff
  48. A Markov trace becomes a knot polynomial
  49. A finite index can leave a boundary in the tunnel
  50. A generating tunnel with incompatible reflected traces
  51. One stabilized square determines the inclusion
  52. Relative hypertraces and finite Følner projections
  53. A common tensor factor in a core inclusion
  54. Transported cups realize a smaller core
  55. Changing a core changes its canonical trace by n²
  56. Bounded frames can keep their central support
  57. Pinching errors and completing supported frames
  58. Finite Fourier bases produce one small quantized corner
  59. Central dimensions and a common quantized corner
  60. Supported frames give local approximation corners
  61. Full support from a factorial larger core
  62. Local corners can approximate the whole inclusion
  63. Approximation can preserve every chosen tunnel prefix
  64. Smooth representations and compression onto the tracial tower
  65. Scalar commutants and corrected tracial reflection
  66. Canonical rescaling of Jones cups
  67. Two-step cups and composed densities
  68. Finite comparisons without a limit map
  69. Canonical density transitivity
  70. Finite trace correction and the orbit of a cup
  71. A common basis bounds both central transitions
  72. Stationarity alone does not control joint localization
  73. Joint projection transfer and central partition flow
  74. Entropy bounds the central partition boundary
  75. Finite branches reduce the joint central test
  76. Variable integer dimensions and central flags
  77. Unequal supports give finite local approximation
  78. General corners and piecewise commuting squares
  79. Whole blocks retaining a prescribed prefix
  80. Relative entropy detects the extremal graph norm
  81. Exact trace certificates close the finite-depth residual
  82. Finite trace order and the remaining support
  83. Central capacity and small support cuts
  84. Relative norm averaging and central density
  85. Finite cup densities and a positive-cost Følner criterion
  86. Normal central hypertraces and joint localization
  87. Finite centers and one common multiplicity
  88. Canonical capacity and optimal dimension repair
  89. Common finite capacity and uniform rounding
  90. Cup-tail commutants and the remaining central comparison
  91. Hypertraces and their normal central part
  92. Finite completion with separate tunnel cells
  93. Singular hypertraces, the Jones ideal and bounded entropy
  94. Selecting a compatible hypertrace
  95. Finite budgets and whole-tunnel operator transfer
  96. Tail supports, fixed operators and compressed tunnel depth
  97. Averaging compatible states and the remaining cost
  98. The invariant cost and common-center variance
  99. General tail supports and controlled tunnels