Open Mathematics Courses

Operator algebras: representations, derivations and core traces

Integrated representations and C*-duality, logarithmic products and power strips, inner derivations, and a direct faithful-state construction of the continuous-core trace.

  1. Continuous calculus, positivity and Hilbert spaces
  2. Scalar measure, convergence and calculus
  3. Positive functionals and nonunital representations
  4. Compact topology and the Hilbert tensor construction
  5. Radon representation, qualified products and Haar measure
  6. Unitary representations and the two group C* completions
  7. Covariance and crossed products with nonunital coefficients
  8. Fourier completion and the character topology
  9. Positive coefficients, the dual measure and scalar Plancherel
  10. Reduced Fourier norms, biduality and the returned measure
  11. Duality for C* crossed products
  12. Operator foundations: spectral domains, normal topology and scalar analysis
  13. Local Fourier and closed-form foundations for the state core
  14. Concrete preduals from Hilbert tensors: the bounded CP-01–06 provider
  15. Bounded strip comparison from scalar proofs
  16. Projection comparison and the countably decomposable type III case
  17. Closed conjugate-linear involutions from their graph Hilbert space
  18. Norm-controlled density on arbitrary Hilbert spaces
  19. Concrete predual balls and faithful ultraweak representations
  20. Bounded vectors, affiliated multipliers and the complete dual Hilbert algebra
  21. Full left-Hilbert-algebra domains and the unchanged closed involution
  22. A weight from a full Hilbert algebra, with its entire finite ideal
  23. A faithful vector-series state has an ultraweak GNS representation
  24. Finite ideals and GNS spaces for arbitrary weights
  25. Order-normal positive functionals and ultraweak continuity
  26. Convex separation and the full dual-ball criterion
  27. Order normality and ultraweak lower semicontinuity for every weight
  28. Extended positive elements and composition with operator-valued weights
  29. The finite-star involution, full algebra and reverse weight correspondence
  30. Faithful normal GNS representations and their topologies
  31. Weight/GNS transport after the forward construction
  32. MF-06: narrow scalar and vector integration bridge
  33. Original proof of the modular fundamental theorem at arbitrary Hilbert-algebra scope
  34. Modular covariance and the canonical opposite weight
  35. A commutant weight without faithfulness or semifiniteness of the input
  36. The KMS strip and a complete analytic finite-star core
  37. Every normal weight is a sum on its whole positive cone
  38. Tracial densities on the complete extended positive cone
  39. Uniqueness of the modular group from the finite-star KMS condition
  40. Balanced matrix weights, exact corner domains and cocycle identities
  41. Entire modular elements and a finite-domain strip criterion
  42. Unbounded centralizer densities: full finite domains and exact cocycles
  43. Natural cones for arbitrary faithful normal semifinite weights
  44. All projection corners and canonical positive-functional vectors
  45. Canonical standard-form transport and finite balanced matrix cones
  46. A periodic compact crossed product and its full counting weight
  47. Real-line Fourier filters, local division and full unitary generators
  48. Weak-star localization for real automorphism actions
  49. Normal spectral transport, typeIII corners and matrix amplification
  50. The direct III1 full invariant-corner spectral bridge
  51. Modular restriction and cocycle preservation for operator-valued weights
  52. The complete compact dual modular group and its scaling trace
  53. Faithful descent through the complete counting operator-valued weight
  54. Discrete crossed products and their compact double dual
  55. Converting a given inner modular period into a periodic weight
  56. The centralizer of a weight with the specified type III parameter period
  57. A spectral unitary and discrete decomposition from an infinite periodic weight
  58. Countable amplification of an arbitrary faithful normal semifinite weight
  59. The real Connes spectrum: orbit supports, corners and cocycles
  60. The full weight spectral identity and the modular intersection comparison
  61. Logarithmic products and bounded perturbations
  62. Power forms and contractive strips
  63. Derivations and uniformly continuous automorphism groups
  64. Inner derivations in an arbitrary von Neumann algebra
  65. From a discrete modular spectral invariant to an actual inner period
  66. Zero in every modular spectrum and the type III spectral trichotomy
  67. Discrete decomposition exists for every separable-predual type III lambda factor
  68. Modular spectral gaps and the minimum distance between commuting functionals
  69. Wide modular gaps and normal states in a type III₀ factor
  70. Faithful normal states: GNS normality and the modular application
  71. The faithful-state core: finite trace vectors, averaging and scaling
  72. Polar cutoffs from finite spectral partitions
  73. Approximate unitary homogeneity of normal states
  74. Normal-state orbit diameter at a positive type III parameter
  75. Given unitary cocycles for arbitrary locally compact groups
  76. Bounded additive cocycles and the small unitary branches
  77. Local Fourier division and closed ideals on an arbitrary LCA group
  78. General normal actions: predual continuity and the integrated maps
  79. Bounded perturbations and differentiability domains of real actions
  80. A normal regular construction on arbitrary Hilbert spaces
  81. Normal-action spectral localization on an arbitrary LCA group
  82. Singleton synthesis and exact eigenoperators on an arbitrary LCA group
  83. Spectral bimodules, invariant support projections and corners
  84. The normal crossed-product commutant for arbitrary locally compact groups
  85. Weyl systems on arbitrary locally compact abelian groups
  86. Spectral calculus for uniformly bounded Banach actions of an arbitrary LCA group
  87. The weak-star generator and its domain
  88. A dual weight from compact coefficient graphs
  89. Local foundations for quotient fields
  90. Quotient measure and the subgroup modular correction
  91. Inducing a representation through a quotient measure
  92. Recovering the subgroup representation from imprimitivity
  93. Why induction preserves every intertwiner
  94. Laplace resolvents of the weak-star generator
  95. A weak-star Hille–Yosida criterion for isometric groups
  96. Whole-cone averaging and comparison of general dual weights
  97. A normal crossed product as a fixed algebra
  98. Inputs for action frequencies and norm continuity
  99. Narrow spectral bands give nearly character orbits
  100. Four tests for an action frequency
  101. The spectrum of one action operator
  102. An isometry with one spectral value
  103. Characters of the filter algebra
  104. Compact frequencies and norm-continuous actions
  105. Norm-continuous groups in a Banach algebra
  106. Stabilization of a normal induced crossed product
  107. Pulling a central quotient back to the group
  108. Recognizing induced actions from a central quotient
  109. Induction and crossed products for discrete groups
  110. Compact averaging identifies the subgroup fixed algebra
  111. Stabilization through compact averaging
  112. Open subgroups and the regular corner
  113. A continuous spectrum for an integrable abelian action
  114. Conditional probabilities in the Haar class of each orbit
  115. A proper topology for an integrable orbit model
  116. Bounded orbit sums need not be continuous
  117. Crossed products over varying compact stabilizers
  118. Conditional measures for arbitrary Borel maps
  119. Decomposing an action over its fixed centre
  120. Conditional measures as ergodic components
  121. Standard forms over a moving central base
  122. Invariant states over an abelian commutant
  123. Covariant representations over a moving diagonal
  124. A suspension from separated crossings
  125. Cohomology reduction for measurable flows
  126. Normal operator filters and their relative topology
  127. Why operator and vector frequencies add
  128. A spectral filtration determines operator frequencies
  129. Two crossed products and the surviving action
  130. Recovering multiplicity from a Weyl system
  131. Fourier densities identify the whole dual-action average
  132. The whole dual-action average and its finite domains
  133. Recovering a weight from a unitary modular cocycle
  134. Recognizing faithful dual weights
  135. Recognizing dual-invariant normal weights, including their infinite part
  136. Supported dual weights: normalization and models
  137. Recovering a crossed product from its eigenunitaries
  138. Haar coordinates and integration after recognition
  139. Tensoring a weight with the usual trace
  140. The second dual weight and its normalization
  141. A spectral coordinate from a trace-scaling action
  142. The core trace and its changes of weight
  143. Ergodic compact actions have a canonical trace
  144. A Gram isometry recovers the compact action
  145. Connes spectrum through fixed corners
  146. Cocycle matrix actions and the representation continuity test
  147. Local displacement and the continuous C-star part
  148. Integrable actions: orbit averages, filters and actual eigenoperators
  149. Lifting innerness and cocycles from a full corner
  150. Extending abelian representations inside their original closure
  151. The dual-center kernel through central overlap
  152. The fixed center and the factor criterion
  153. The normalized trace with its full center retained
  154. Trace scaling, central corners and the type III criterion
  155. Recovering a continuous core with its specified trace
  156. Continuous decompositions of arbitrary type III algebras
  157. Supported weights and their coordinates in the continuous core
  158. Graded fibers and changing weight coordinates
  159. Integrating graded sections
  160. Hilbert fibers and the represented core
  161. Countable orbit algebras and their modular weight
  162. Lifted orbit relations and the normalized core
  163. Cancel an inner subgroup, then detect the exact spectrum
  164. Transport a fixed corner, then assemble its local cardinal
  165. Modular spectra and periods from the center flow
  166. Canonical core automorphisms and their continuity
  167. Relative commutants and central cocycles
  168. Dominant weights and inner comparison
  169. Trace-scaling cocycles and fixed corners
  170. Mutual corner approximation through a family of spectral bridges
  171. Small fixed corners, inner spectra and annihilating times
  172. From an intertwiner family to tensor-product innerness
  173. Read an inner implementer from a circle eigenunitary
  174. A central order correction fixes the implementer and retains its bound
  175. Compact spectral images and central fixed implementers
  176. Read the scalar phase, normalize, and obtain the exact kernel
  177. Discrete Fourier columns, finite corners and orbit MASAs
  178. Finite free actions and cocycle stability
  179. Internal Hilbert spaces in a von Neumann algebra
  180. Operator spaces and endomorphisms from internal Hilbert spaces
  181. Internal representations and irreducible multipliers
  182. Compact Tannaka reconstruction and centralizer rigidity

Prerequisite proofs · Results and dependencies

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