Measured relations and operator algebras

These lessons begin with orbit relations, then study diagonal-preserving operators, invariant measures, finite classes, isotropy, and amenable approximation.

  1. Orbits, stabilizers, and relation algebras
  2. Measurable actions and compact models
  3. Free actions and the crossed-product diagonal
  4. Type criteria for locally compact free actions
  5. Relation kernels and modular coordinates
  6. Ratio sets and intrinsic modular spectra
  7. Normalizers, phases, and orbit cocycles
  8. Diagonal expectations and invariant measures
  9. Lifted relations and the associated flow
  10. Polish orbits and their quotient topology
  11. Locally closed orbits and measurable representatives
  12. Groupoids and measured orbit relations
  13. Finite orbit classes and matrix blocks
  14. Isotropy and random-operator fibres
  15. Semifinite transverse measures and operator completions
  16. Countable generation and isotropy topologies
  17. Borel group measures and isotropy topologies
  18. Averaged coefficients and isotropy commutants
  19. Modular orbit integrals and spectral coordinates
  20. Orbit averaging and the modular weight bridge
  21. Joint spectral charts and measurable intertwiners
  22. Almost homomorphisms on measured groupoids
  23. Strict spectral representations on the stable kernel
  24. Integrable centralizers and spectral intertwiners
  25. Spectral necessity and modular transfer
  26. Commuting copies in principal groupoid factors
  27. Ergodic transverse measures and extremal rays
  28. Principal groupoids with extra fibre information
  29. Principal groupoids with hidden group factors
  30. Means, Følner sets, and regular representations
  31. Haar averages and compact translation control
  32. Closed subgroups and continuous averaging
  33. Almost-connected groups and the solvable radical
  34. Invariant means on measured relations
  35. Towers and odometer orbits
  36. Compatible lifts and cohomology reduction
  37. Kernels and measurable groupoid splitting
  38. Asymptotic ranges and groupoid cohomology
  39. Balanced arrays and the hyperfinite finite factor
  40. Matching sets and nonsingular dyadic arrays
  41. Ancillary actions and unitary corrections
  42. Localizing factor actions and uniform cocycles
  43. Return arrows and factor-field flows
  44. Stabilizer fields and ancillary splittings
  45. Variable factor fields and measurable conjugacy
  46. Strict variable fields and ancillary conjugacy
  47. Dense roof groups, two ceilings and eigenfunctions
  48. Cocycle groupoids and semidirect products
  49. Type I stages in irrational rotation factors
  50. Amenable actions, free centres, and isotropy obstructions
  51. Compact fixed sets and free Polish models
  52. Orbit representatives and null fibre exceptions
  53. Fourier cutoffs and the free diagonal
  54. Weyl lattices and coupling dimension
  55. Borel carriers and uniform wandering neighborhoods
  56. Separable compact actions with nonregular measures
  57. From actions to orbit algebras