A smooth version of Johnson's problem on derivations of group algebras

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We give a description of the algebra of outer derivations of the group algebra of a finitely presented discrete group in terms of the Cayley complex of the groupoid of the adjoint action of the group. This problem is a smooth version of Johnson's problem on derivations of a group algebra. We show that the algebra of outer derivations is isomorphic to the one-dimensional compactly supported cohomology group of the Cayley complex over the field of complex numbers. Bibliography: 34 titles.

Sobre autores

Andronick Arutyunov

Moscow Institute of Physics and Technology (State University)

Email: andronick.arutyunov@gmail.com
Candidate of physico-mathematical sciences, no status

Alexandr Mishchenko

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Email: asmish@mech.math.msu.su
Doctor of physico-mathematical sciences, Professor

Bibliografia

  1. B. E. Johnson, A. M. Sinclair, “Continuity of derivations and a problem of Kaplansky”, Amer. J. Math., 90:4 (1968), 1067–1073
  2. B. E. Johnson, J. R. Ringrose, “Derivations of operator algebras and discrete group algebras”, Bull. London Math. Soc., 1 (1969), 70–74
  3. B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc., 127, Amer. Math. Soc., Providence, RI, 1972, iii+96 pp.
  4. B. E. Johnson, “The derivation problem for group algebras of connected locally compact groups”, J. London Math. Soc. (2), 63:2 (2001), 441–452
  5. V. Losert, “The derivation problem for group algebras”, Ann. of Math. (2), 168:1 (2008), 221–246
  6. H. G. Dales, Banach algebras and automatic continuity, London Math. Soc. Monogr. (N.S.), 24, The Clarendon Press, Oxford Univ. Press, New York, 2000, xviii+907 pp.
  7. Р. Пирс, Ассоциативные алгебры, Мир, М., 1986, 543 с.
  8. C. A. Weibel, An introduction to homological algebra, Cambridge Stud. Adv. Math., 38, Reprint, Cambridge Univ. Press, Cambridge, 1997, xiv+450 pp.
  9. I. Kaplansky, “Modules over operator algebras”, Amer. J. Math., 75:4 (1953), 839–858
  10. I. Kaplansky, “Derivations of Banach algebras”, Seminars on analytic functions, v. II, Princeton Univ. Press, Princeton, N.J., 1958, 254–258
  11. S. Sakai, “On a conjecture of Kaplansky”, Tôhoku Math. J. (2), 12 (1960), 31–33
  12. S. Sakai, “Derivations of $W^{*}$-algebras”, Ann. of Math. (2), 83 (1966), 273–279
  13. S. Sakai, “Derivations of simple $C^{*}$-algebras”, J. Functional Analysis, 2:2 (1968), 202–206
  14. S. Sakai, $C^{*}$-algebras and $W^{*}$-algebras, Ergeb. Math. Grenzgeb., 60, Springer-Verlag, New York–Heidelberg, 1971, xii+253 pp.
  15. R. V. Kadison, “Derivations of operator algebras”, Ann. of Math. (2), 83:2 (1966), 280–293
  16. R. V. Kadison, J. R. Ringrose, “Derivations of operator group algebras”, Amer. J. Math., 88:3 (1966), 562–576
  17. B. Blackadar, J. Cuntz, “Differential Banach algebra norms and smooth subalgebras of $C^*$-algebras”, J. Operator Theory, 26:2 (1991), 255–282
  18. V. Ginzburg, Letures on noncommutative geometry
  19. A. Connes, “Non-commutative differential geometry”, Inst. Hautes Etudes Sci. Publ. Math., 62 (1985), 41–144
  20. A. Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994, xiv+661 pp.
  21. A. Connes, H. Moscovici, “Cyclic cohomology, the Novikov conjecture and hyperbolic groups”, Topology, 29:3 (1990), 345–388
  22. P. Jolissaint, “Rapidly decreasing functions in reduced $C^*$-algebras of groups”, Trans. Amer. Math. Soc., 317:1 (1990), 167–196
  23. P. Jolissaint, “$K$-theory of reduced $C^*$-algebras and rapidly decreasing functions on groups”, K-Theory, 2:6 (1989), 723–735
  24. P. de la Harpe, “Groupes hyperboliques, algèbres d'operateurs et un theorème de Jolissaint”, C. R. Acad. Sci. Paris Ser. I Math., 307:14 (1988), 771–774
  25. А. В. Ершов, Категории и функторы, Учебное пособие, Наука, Саратов, 2012, 86 с.
  26. Р. Линдон, П. Шупп, Комбинаторная теория групп, Мир, М., 1980, 448 с.
  27. М. И. Каргаполов, Ю. И. Мерзляков, Основы теории групп, 3-е изд., Наука, М., 1982, 288 с.
  28. В. Магнус, А. Каррас, Д. Солитэр, Комбинаторная теория групп. Представление групп в терминах образующих и соотношений, Наука, М., 1974, 455 с.
  29. D. J. Benson, Representations and cohomology, v. I, Cambridge Stud. Adv. Math., 30, Basic representation theory of finite groups and associative algebras, Cambridge Univ. Press, 1991, xii+224 pp.
  30. M. Gerstenhaber, “The cohomology structure of an associative ring”, Ann. of Math. (2), 78:2 (1963), 267–288
  31. M. Kontsevich, “Operads and motives in deformation quantization”, Lett. Math. Phys., 48:1 (1999), 35–72
  32. Y. Felix, J.-Cl. Thomas, M. Vigue-Poirrier, “The Hochschild cohomology of a closed manifold”, Publ. Math. Inst. Hautes Etudes Sci., 99 (2004), 235–252
  33. А. А. Арутюнов, А. С. Мищенко, А. И. Штерн, “Деривации групповых алгебр”, Фундамент. и прикл. матем., 21:6 (2016), 65–78
  34. A. A. Arutyunov, A. S. Mishchenko, Smooth version of Johnson's problem concerning derivations of group algebras

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