Open Mathematics Courses

Operator algebras: crossed products, spectra and flows

Operator-algebra foundations, crossed products and dual weights, action spectra and derivations, continuous and discrete decompositions, and canonical cores.

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. Modular covariance through positive multipliers and operator averages
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. Scalar continuity implies norm continuity for Banach representations
79. General normal actions: predual continuity and the integrated maps
80. Bounded perturbations and differentiability domains of real actions
81. A normal regular construction on arbitrary Hilbert spaces
82. Normal-action spectral localization on an arbitrary LCA group
83. Singleton synthesis and exact eigenoperators on an arbitrary LCA group
84. Spectral bimodules, invariant support projections and corners
85. Compact coefficient integration for normal covariant representations
86. Weyl systems on arbitrary locally compact abelian groups
87. Spectral calculus for uniformly bounded Banach actions of an arbitrary LCA group
88. Generators and their norm and weak-star domains
89. A dual weight from compact coefficient graphs
90. The normal crossed-product commutant for arbitrary locally compact groups
91. Local foundations for quotient fields
92. Quotient measure and the subgroup modular correction
93. Inducing a representation through a quotient measure
94. Recovering the subgroup representation from imprimitivity
95. Why induction preserves every intertwiner
96. Laplace resolvents and full generator domains
97. Isometric group generation in norm and weak-star topologies
98. Whole-cone averaging and comparison of general dual weights
99. A normal crossed product as a fixed algebra
100. Inputs for action frequencies and norm continuity
101. Narrow spectral bands give nearly character orbits
102. Four tests for an action frequency
103. The spectrum of one action operator
104. An isometry with one spectral value
105. Characters of the filter algebra
106. Compact frequencies and norm-continuous actions
107. Norm-continuous groups in a Banach algebra
108. Modulation, reciprocal tails and the full Fourier generator
109. Closed-set Fourier quotients and their full generators
110. The spectrum of the full generator of a real action
111. Invariant weights realize the action spectrum
112. Strongly continuous orbits and uniform continuity
113. Spectral fibers of circle translation
114. Faithful ergodic actions of the circle
115. Compact translations and twisted torus actions
116. Stabilization of a normal induced crossed product
117. Pulling a central quotient back to the group
118. Recognizing induced actions from a central quotient
119. Induction and crossed products for discrete groups
120. Compact averaging identifies the subgroup fixed algebra
121. Stabilization through compact averaging
122. Open subgroups and the regular corner
123. A continuous spectrum for an integrable abelian action
124. Integrability makes an ergodic abelian action homogeneous
125. Conditional probabilities in the Haar class of each orbit
126. A proper topology for an integrable orbit model
127. Bounded orbit sums need not be continuous
128. Crossed products over varying compact stabilizers
129. Conditional measures for arbitrary Borel maps
130. Decomposing an action over its fixed centre
131. Conditional measures as ergodic components
132. Standard forms over a moving central base
133. Measurable standard forms and canonical transport between varying fibres
134. Invariant states over an abelian commutant
135. Covariant representations over a moving diagonal
136. A suspension from separated crossings
137. Cohomology reduction for measurable flows
138. Normal operator filters and their relative topology
139. Why operator and vector frequencies add
140. A spectral filtration determines operator frequencies
141. Positive implementation from spectral tails
142. Two crossed products and the surviving action
143. Recovering multiplicity from a Weyl system
144. Fourier densities identify the whole dual-action average
145. The whole dual-action average and its finite domains
146. Recovering a weight from a unitary modular cocycle
147. Recognizing faithful dual weights
148. Recognizing dual-invariant normal weights, including their infinite part
149. Supported dual weights: normalization and models
150. Recovering a crossed product from its eigenunitaries
151. Haar coordinates and integration after recognition
152. Tensoring a weight with the usual trace
153. The second dual weight and its normalization
154. A spectral coordinate from a trace-scaling action
155. The core trace and its changes of weight
156. Ergodic compact actions have a canonical trace
157. A Gram isometry recovers the compact action
158. Spectral unitaries from twisted matrix fixed points
159. Fourier modules and spectral fibers in crossed products
160. Spectral transfer on ordinary Banach spaces
161. Connes spectrum through fixed corners
162. Cocycle matrix actions and the representation continuity test
163. Local displacement and the continuous C-star part
164. Integrable actions: orbit averages, filters and actual eigenoperators
165. Lifting innerness and cocycles from a full corner
166. Extending abelian representations inside their original closure
167. The dual-center kernel through central overlap
168. The fixed center and the factor criterion
169. The normalized trace with its full center retained
170. Trace scaling, central corners and the type III criterion
171. Recovering a continuous core with its specified trace
172. Continuous decompositions of arbitrary type III algebras
173. Supported weights and their coordinates in the continuous core
174. Graded fibers and changing weight coordinates
175. Integrating graded sections
176. Hilbert fibers and the represented core
177. Countable orbit algebras and their modular weight
178. Lifted orbit relations and the normalized core
179. Cancel an inner subgroup, then detect the exact spectrum
180. Transport a fixed corner, then assemble its local cardinal
181. Modular spectra, periods and almost-periodic states
182. Canonical core automorphisms and their continuity
183. Relative commutants and central cocycles
184. Dominant weights and inner comparison
185. Trace-scaling cocycles and fixed corners
186. Automorphisms and the outer exact sequence
187. Pointwise absorption and dominant weights
188. Regular intertwiners and integrable cocycles
189. The global carrier and integrable weights
190. Translation cocycles and the semifinite core
191. Integrable weights and type-I densities
192. From the global carrier to the fixed-trace density
193. Finite carrier projections and their normal center measures
194. Discrete decomposition and periodic weights
195. Periodic weights and the scaling modulus
196. Tensor products and the normalized flow of weights
197. From a central cocycle to a full return algebra
198. Type III zero factors from a contracting return automorphism
199. Entropy at return times
200. Changing the return section without changing the crossed product
201. The same hyperfinite factor from different return systems
202. Norm smoothing of measurable scalar cocycles
203. Krieger's tower and its stopping invariant
204. Mutual corner approximation through a family of spectral bridges
205. Small fixed corners, inner spectra and annihilating times
206. From an intertwiner family to tensor-product innerness
207. Read an inner implementer from a circle eigenunitary
208. A central order correction fixes the implementer and retains its bound
209. Compact spectral images and central fixed implementers
210. Read the scalar phase, normalize, and obtain the exact kernel
211. Discrete Fourier columns, finite corners and orbit MASAs
212. Spectral normalizers and irrational rotation factors
213. Recovering a discrete system from a modular spectral gap
214. Tensor periods, modular subtypes and absorption
215. Central perturbations, spectral gaps, and ambient maximal abelian algebras
216. Spectral fibers and the full carrier of type I weights
217. Normalized weights in a discrete coefficient algebra
218. A fundamental band for the global weight carrier
219. States with abelian centralizers and mutual commutants
220. Finite free actions and cocycle stability
221. Internal Hilbert spaces in a von Neumann algebra
222. Operator spaces and endomorphisms from internal Hilbert spaces
223. Internal representations and irreducible multipliers
224. Compact Tannaka reconstruction and centralizer rigidity
225. The Weyl-plane action and its continuous part
226. Principal logarithms and small automorphisms
227. Pointwise inner flows and continuous implementers
228. Multiplier innerness, von Neumann uniformity, analytic measures, and positive energy
229. Comparing weights through strips and spectral tails
230. Uniform distance and limits of weights
231. Commuting approximations and finite domains of weights
232. Trace tails and the smallest common majorant
233. Spectral distributions and closed unitary orbits in semifinite factors
234. A finite spectral cut of a core density
235. Finite orbit blocks and type II∞ approximations of a type III₀ factor
236. Assemble trace tails over the center
237. Compatible expectations carry the core-center invariant to a limit
238. From a compact spectral profile to the continuous core
239. The central functional classifies closed unitary orbits
240. Trace cutoffs, transport and homogeneous domains
241. Finite densities and the graded integral
242. Products, analytic powers and exact dual partners
243. Spectral coordinates, localizers and finite mass
244. A cocycle section needs a genuine joint coordinate
245. Exact changes of weight and the limit of spectral continuity
246. Normalizer defects and the characteristic square
247. Periodic cores and the lifting obstruction
248. Coordinates for equivariant central extensions
249. When measurable coordinates recover a topology
250. Compatible lifts and the phase of an outer period
251. Twisted regular products and prescribed inner kernels
252. Nontracial Bernoulli shifts on the original group of sites
253. Two prescribed gaps on every orbit of a Borel flow
254. Amenable regular products and prescribed inner kernels
255. An irrational characteristic class with no AFD realization

Prerequisite proofs · Results and dependencies

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