Positive maps and finite-dimensional approximation · Open Mathematics Courses

Sources, authorship and component terms

This graduate course develops completely positive maps, matrix approximation, nuclearity, infinite tensor products, central sequences, MASAs and the injectivity/AFD equivalence. It contains fifty lessons, 327 numbered results and 504 exercises with complete solutions.

The reading list credits the research articles and notes used in the exposition. The result register identifies the associated arguments.

Licences

The current course exposition, generated reader and Finite matrix-state foundations, positive-cover measure construction and compactness and measure completion are dedicated under CC0 1.0.

Free access to a paper does not grant permission to copy its expression. Research sources retain their own rights. Their PDFs, scans and working extracts are not included in this edition.

Prerequisite proofs and reading order · Source versions and selected reading scopes · Result comparison register · Download the reader and editable sources

Freely accessible research readings

  1. A note on relative amenability of finite von Neumann algebras — Xiaoyan Zhou, Junsheng Fang. Journal of Operator Theory 81:1 (2019), 107–132.
  2. An introduction to II₁ factors — Claire Anantharaman, Sorin Popa
  3. Inductive limits of Banach algebras — Matthew Daws
  4. A short proof of “injectivity implies hyperfiniteness” for finite von Neumann algebras — Sorin Popa
  5. On a problem of R. V. Kadison on maximal abelian *-subalgebras in factors — Sorin Popa
  6. Outer conjugacy classes of automorphisms of factors — Alain Connes
  7. Subalgebras of C*-algebras — William B. Arveson. Acta Mathematica 123 (1969), 141–224
  8. A theorem on rings of operators — Irving Kaplansky. Pacific Journal of Mathematics 1(2) (1951), 227–232
  9. Bures Distance Function and a Generalization of Sakai’s Non-commutative Radon–Nikodym Theorem — Huzihiro Araki. Publications of the Research Institute for Mathematical Sciences 8 (1972/73), 335–362; visible journal scan,28 PDF pages; publisher year metadata1972
  10. An extension of Kakutani’s theorem on infinite product measures to the tensor product of semifinite W*-algebras — Donald J. C. Bures. Transactions of the American Mathematical Society 135 (1969), 199–212; exact14-page publisher scan
  11. The standard form of von Neumann algebras — Uffe Haagerup. Mathematica Scandinavica 37 (1975), 271–283
  12. Some properties of modular conjugation operator of von Neumann algebras and a non-commutative Radon–Nikodym theorem with a chain rule — Huzihiro Araki
  13. The symbiosis of C*- and W*-algebras — Nathanial P. Brown. arXiv:0812.1763v1, 9 December 2008; only revision shown in the actually read submission history
  14. Concise lectures on selected topics of von Neumann algebras — Fumio Hiai. arXiv:2004.02383v1, 6 April 2020 Cone duality and the positive modular correction in Section3.
  15. On the dual weights for crossed products of von Neumann algebras I: Removing separability conditions — Uffe Haagerup. Mathematica Scandinavica 43 (1978), 99–118, DOI 10.7146/math.scand.a-11768
  16. The Connes Embedding Problem: Sofic groups and the QWEP Conjecture — Kristian Knudsen Olesen. University of Copenhagen master’s thesis, November 2012; Section 2.4, pp.51–56.
  17. Lifting theorems for completely positive maps — James Gabe. Accepted arXiv:1508.00389v4, 30 January 2022
  18. Actions of discrete amenable groups on factors — Adrian Ocneanu. University of Warwick PhD thesis, submitted February 1982; WRAP submitted-version scan
  19. Ultraproducts of von Neumann algebras — Hiroshi Ando, Uffe Haagerup
  20. On the structure of II₁-factors — Dusa McDuff. Russian Mathematical Surveys 25(6) (1970), 29–50; English version supplied by MathNet; DOI 10.1070/RM1970v025n06ABEH001267
  21. Automatic continuity of group homomorphisms — Christian Rosendal
  22. The dual space of an operator algebra — Charles A. Akemann
  23. Grothendieck's Theorem, past and present — Gilles Pisier. Expanded UNCUT author version, cover dated 21 August 2013, 93 PDF pages; distinct from the 2012 published survey and from arXiv v3.
  24. Grothendieck's Theorem, past and present — Gilles Pisier
  25. On matricially normed spaces — Edward G. Effros, Zhong-Jin Ruan
  26. The Functional Analysis of Quantum Information Theory — Ved Prakash Gupta, Prabha Mandayam, V. S. Sunder
  27. On simplicity of reduced C*-algebras of groups — Pierre de la Harpe
  28. The trace on the K-theory of group C*-algebras — Thomas Schick
  29. K-groups of reduced crossed products by free groups — Mihai Pimsner, Dan Voiculescu
  30. A Classification of Factors — Huzihiro Araki, E. J. Woods. Actual scanned cover says Publ. RIMS Kyoto Univ. 3 (1968), 51–130. Official EMS article 2441 indexes volume 4 (1968), no.1, published 30 April 1968. These literal, differing volume fields are retained; no inferred correction of the cover.
  31. On Quasifree States of CAR and Bogoliubov Automorphisms — Huzihiro Araki. Printed header volume 6 (1970/71), 385–442; received 6 May and revised 17 November 1970. Official EMS article 2511 indexes volume 6 (1970), no.3, published 31 December 1970. Actual 58-page publisher scan.
  32. Introduction to representations of the canonical commutation and anticommutation relations — Jan Dereziński
  33. On infinite direct products — J. von Neumann. Compositio Mathematica 6 (1939), 1–77; Numdam wrapper plus 77 printed pages, actual 78-page PDF. The year 1939 follows the original wrapper, not Nakagami reference 11, which prints 1937.
  34. Inner automorphisms of hyperfinite factors and Bogoliubov transformations — A. L. Carey. Annales de l’Institut Henri Poincaré, Physique théorique 40(2) (1984), 141–149; actual 10-page Numdam PDF including wrapper.
  35. Infinite Tensor Products of Operators — Yoshiomi Nakagami. Publications of the Research Institute for Mathematical Sciences 10 (1974), 111–145; communicated 19 October 1973. Actual 36-page publisher PDF including final blank page; note added in proof on printed p.145 retained.
  36. Automorphic group representations — Robert J. Blattner. Pacific Journal of Mathematics 8(4) (1958), 665–677; actual publisher PDF has 17 pages including cover and journal matter. The actual cover prints June 1958; received 10 April 1958.
  37. The Pukánszky invariant for masas in group von Neumann factors — Allan M. Sinclair, Roger R. Smith. Undated 25-page author-hosted manuscript; corresponding article Illinois Journal of Mathematics 49 (2005), 325-343, verified from author publication list. PDF CreationDate is 2007-11-06 and is a file-production date, not publication date. Manuscript printed page equals one-based PDF page.
  38. Orthogonal pairs of subalgebras in finite factors — Sorin Popa. INCREST preprint 89/1981, title leaf explicitly October 1981; 14-page scanned file produced in 2017. Printed page p equals PDF page p+4 for the proof pages used.
  39. An amenable equivalence relation is generated by a single transformation — Alain Connes, Jacob Feldman, Benjamin Weiss. Ergodic Theory and Dynamical Systems 1(4) (1981), 431–450; actual Cambridge publisher PDF, 20 pages
  40. The equivalence of various definitions for a properly infinite von Neumann algebra to be approximately finite dimensional — George A. Elliott, E. J. Woods. Published Proceedings of the American Mathematical Society 60, October 1976, printed pp.175–178; actual PDF prints copyright 1977. Four-page AMS publisher version.
  41. On Maximal Abelian Subrings of Factors of Type II₁ — Lajos Pukánszky
  42. Finite Fields — Keith Conrad
  43. MIT 18.781 Spring 2012 Lecture 12: Cyclotomic Polynomials, Primes Congruent to 1 mod n — Abhinav Kumar. Abhinav Kumar, MIT18.781 Theory of Numbers, Spring2012, Lecture12, Cyclotomic Polynomials, Primes Congruent to1modn. Official MITOCW PDF metadata Author Kumar, Abhinav; course page independently identifies Kumar and Spring2012. Five PDF pages, four numbered content pages followed by MITOCW attribution leaf. Printed and PDF page numbers agree for content pages.
  44. On approximately finite-dimensional von Neumann algebras — George A. Elliott. Published scan, Math. Scand. 39 (1976), 91–101; received June 12, 1975; footnotes added June 18, 1976.
  45. On approximately finite-dimensional von Neuman algebras, II — George A. Elliott
  46. On the projection of norm one in W*-algebras — Jun Tomiyama
  47. Periodic automorphisms of the hyperfinite factor of type II1 — Alain Connes
  48. Math 261y: von Neumann Algebras, Lecture 18 — Jacob Lurie
  49. Math 261y: von Neumann Algebras, Lecture 19 — Jacob Lurie
  50. Math 261y: von Neumann Algebras, Lecture 20 — Jacob Lurie
  51. Introduction to Descriptive Set Theory — Anush Tserunyan
  52. Selected results on measurable selections — Siegfried Graf
  53. Notes on operator algebras — Jesse Peterson
  54. Functional Analysis, Princeton University MAT520 Lecture Notes — Jacob Shapiro
  55. Measure, Integration & Real Analysis — Sheldon Axler. Author-hosted revision of 12 June 2026; first edition Springer GTM 282 (2020)
  56. Math 261y: von Neumann Algebras (Lecture 11) — Jacob Lurie. September 26, 2011
  57. Math 261y: von Neumann Algebras (Lecture 12) — Jacob Lurie. September 27, 2011
  58. A new proof of the existence of a trace in a finite von Neumann algebra — F. J. Yeadon. Bulletin of the American Mathematical Society 77 (1971), 257–260
  59. Formes linéaires sur un anneau d'opérateurs — Jacques Dixmier. Bulletin de la Société Mathématique de France 81 (1953), 9–39
  60. Stats 310A (Math 230A) Lecture notes — Sourav Chatterjee