Open Mathematics Courses

Modular theory and weights

Weights, semicyclic representations and the Tomita–Takesaki theory: finite domains, standard forms, modular groups and KMS states, Connes cocycles, operator-valued weights, noncommutative integration and the Haagerup and Connes–Takesaki constructions.

136 lessons.

  1. Modular theory through weights, time and reconstruction
  2. Finite domains and representations of general weights
  3. Weights and semicyclic representations: the exact opening package
  4. Comparing weights through finite-energy vectors
  5. A bounded-operator kernel for weight arguments
  6. Finite domains, null directions, and support corners
  7. Concrete preduals from Hilbert tensors
  8. Detecting normal weights by finite observations
  9. The predual-valued map of a normal weight
  10. Countable amplification and ultraweak bicommutant closure
  11. Normal positive maps and their preadjoints
  12. The vector retained by a comparison map
  13. A faithful weight on the GNS commutant
  14. Spectral calculus with its domains retained
  15. Banach holomorphy, generators, and resolvent limits
  16. Recovering operators from energy forms
  17. Closing energy domains and comparing resolvents
  18. Compatible pairs and complex interpolation
  19. Closing an involution and recovering its modular data
  20. Building the two multiplication actions of a Hilbert algebra
  21. Haar convolution as left and right Hilbert algebras
  22. Polynomial cutoffs recover an operator's range
  23. Right Hilbert algebra density and commutants
  24. The full left Hilbert algebra obtained by dualizing twice
  25. Weights and the Hilbert spaces of multiplication
  26. Changing algebra coordinates and scaling a weight
  27. Analytic kernels for unbounded modular operators
  28. Approximating multiplication without losing its bound
  29. The modular group and its analytic algebra
  30. The positive cone of a standard representation
  31. Recovering a representation from its positive cone
  32. Order and concavity of positive implementing vectors
  33. The modulus and phase of a vector
  34. Positive vectors at half and quarter modular time
  35. A bounded algebra inside its positive cone
  36. Recovering Jordan multiplication from the positive cone
  37. Implementing Jordan symmetries on standard forms
  38. Positive changes of scale on a standard cone
  39. Inner implementers from spectral tails
  40. Bounded generators of cone symmetries
  41. Why an everywhere-defined derivation is bounded
  42. Recovering an algebra from an oriented cone
  43. Cone symmetries without a global faithful state
  44. Positivity seen through two components
  45. Unbounded products in the standard positive cone
  46. Canonical L2 and standard implementations
  47. Adding square roots of normal functionals
  48. Supported GNS representations and the balanced cocycle
  49. Contractive retractions and conditional expectations
  50. Spatial comparison on an arbitrary representation
  51. Constructing spatial energy from finite observations
  52. Adding spatial energies and changing the reference weight
  53. Spatial energy and modular time
  54. The KMS boundary condition determines the modular group
  55. From a C*-modular condition to the GNS von Neumann algebra
  56. Closed graphs and bounded operator cutoffs
  57. Operators recovered from small trace defects
  58. When a small trace defect must vanish
  59. What a functional can detect in measure
  60. Functions, cutoffs and actual domains
  61. Convergence on large domains
  62. Spectral tails, singular values and trace order
  63. Measurable operators between representations
  64. Cyclic families and countable detection
  65. Closed operators through their graph projections
  66. Finite trace domains, central parts and amplification
  67. Averaging, central traces and normality
  68. Relative tensor products and fusion
  69. Fusion over a common spectral point
  70. Reconstructing a weight from a modular cocycle
  71. Recovering a weight from spatial energy
  72. Fixed observations, density weights and modular time
  73. Tracial multiplication actions and the preserving expectation
  74. Conditional expectations from modular invariance
  75. Corner weights tested on all positive energies
  76. Trace densities and noncommutative integration
  77. Spectral layers in a semifinite trace algebra
  78. Recognizing invariant weights by their densities
  79. Inner modular flow and rigidity of finite weight data
  80. Weight domination and exact half-strip endpoints
  81. Partial cocycles and their exact mixed KMS domains
  82. Analytic generators and exact finite weight domains
  83. Upper imaginary time and finite weight multipliers
  84. Commuting modular flows and relative half-strip multipliers
  85. Central cocycles and common finite domains
  86. Normal-functional polar decomposition and invariant pairs
  87. Finite corners, faithful traces, and affine group fixed points
  88. Ergodic factors and trace normalization
  89. Semifinite corners and bounded modular operators
  90. Modular partitions and the full finite domain
  91. Tensor operators and tensor weights
  92. Compatible measurable GNS fields and both representation transports
  93. Measurable weight fields and their integrals
  94. The realization boundary in measurable weight fields
  95. Separable GNS spaces and disintegration of C*-weights
  96. Changing the reference of a weight cocycle
  97. Compact-time continuity along automorphism orbits
  98. Norm continuity of relative modular time
  99. Forced C*-unitization
  100. Finite weight domains and GNS
  101. Bounded functionals and cyclic vectors
  102. States and the positive spanning family
  103. Weak compactness and second adjoints
  104. The universal C*-bidual
  105. Detecting a von Neumann algebra through a separable subalgebra
  106. Lower semicontinuous C*-weights
  107. A supported inverse weight cocycle
  108. Comparing supported weights and transporting their cuts
  109. Uniform geometry of faithful weights
  110. Homogeneous operators in the continuous core
  111. When countable sums detect extended positive energy
  112. Finite calculus and composition of operator-valued weights
  113. Detecting and determining operator-valued weights
  114. Modular invariants of operator-valued weights
  115. Commutant duality for operator-valued weights
  116. Compatible modular weights and operator-valued reconstruction
  117. Countable modular sums and infinite-weight boundaries
  118. Joint spectral measures and commutator estimates in standard form
  119. Measurable Hilbert fields and their diagonal commutant
  120. Central decomposition from countable operator equations
  121. Countable cores, multiplier domains and measurable choices
  122. Recovering algebras from measurable fibres
  123. Regular representations, Fourier algebras, and Borel group measures
  124. Six laboratories on expectations and operator-valued averages
  125. Compatible traces and the coupling operator
  126. A module, its standard corner and its double
  127. Traces on modules and full correspondences
  128. Nontracial Lp spaces for arbitrary von Neumann algebras
  129. Full Hilbert-algebra norms and weighted rank-one models
  130. Closed GNS graphs, cyclic vectors and bounded modular time
  131. The geometry of KMS states and their central densities
  132. Measures on a compact convex space
  133. Central decompositions and disjoint equilibrium representations
  134. Finite pieces and invariant observations for modular flows
  135. Modular time from a faithful normal state
  136. Positive vectors from a faithful state

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