Where to find the image of a derivation

Martin Mathieu

Banach Center Publications (1994)

  • Volume: 30, Issue: 1, page 237-249
  • ISSN: 0137-6934

Abstract

top
With this paper, we intend to provide an overview of some recent work on a problem on unbounded derivations of Banach algebras that still defies solution, the non-commutative Singer-Wermer conjecture. In particular, we discuss several global as well as local properties of derivations entailing quasinilpotency in the image.

How to cite

top

Mathieu, Martin. "Where to find the image of a derivation." Banach Center Publications 30.1 (1994): 237-249. <http://eudml.org/doc/262681>.

@article{Mathieu1994,
abstract = {With this paper, we intend to provide an overview of some recent work on a problem on unbounded derivations of Banach algebras that still defies solution, the non-commutative Singer-Wermer conjecture. In particular, we discuss several global as well as local properties of derivations entailing quasinilpotency in the image.},
author = {Mathieu, Martin},
journal = {Banach Center Publications},
keywords = {unbounded derivations of Banach algebras; non-commutative Singer-Wermer conjecture; quasinilpotency in the image},
language = {eng},
number = {1},
pages = {237-249},
title = {Where to find the image of a derivation},
url = {http://eudml.org/doc/262681},
volume = {30},
year = {1994},
}

TY - JOUR
AU - Mathieu, Martin
TI - Where to find the image of a derivation
JO - Banach Center Publications
PY - 1994
VL - 30
IS - 1
SP - 237
EP - 249
AB - With this paper, we intend to provide an overview of some recent work on a problem on unbounded derivations of Banach algebras that still defies solution, the non-commutative Singer-Wermer conjecture. In particular, we discuss several global as well as local properties of derivations entailing quasinilpotency in the image.
LA - eng
KW - unbounded derivations of Banach algebras; non-commutative Singer-Wermer conjecture; quasinilpotency in the image
UR - http://eudml.org/doc/262681
ER -

References

top
  1. [1] C. Apostol, Inner derivations with closed range, Rev. Roumaine Math. Pures Appl. 21 (1976), 242-265. 
  2. [2] C. Apostol and J. G. Stampfli, On derivation ranges, Indiana Univ. Math. J. 25 (1976), 857-869. Zbl0355.47025
  3. [3] J. Bergen, I. N. Herstein and C. Lanski, Derivations with invertible values, Canad. J. Math. 35 (1983), 300-310. Zbl0522.16031
  4. [4] M. Brešar, Centralizing mappings on von Neumann algebras, Proc. Amer. Math. Soc. 111 (1991), 501-510. Zbl0746.46054
  5. [5] M. Brešar, On a generalization of the notion of centralizing mappings, ibid. 114 (1992), 641-649. Zbl0754.16020
  6. [6] M. Brešar, Derivations decreasing the spectral radius, Arch. Math. (Basel) 61 (1993), 160-162. Zbl0818.46049
  7. [7] M. Brešar and J. Vukman, On left derivations and related mappings, Proc. Amer. Math. Soc. 110 (1990), 7-16. Zbl0703.16020
  8. [8] M. Brešar and J. Vukman, Derivations on noncommutative Banach algebras, Arch. Math. (Basel) 59 (1992), 363-370. Zbl0807.46049
  9. [9] C.-L. Chuang and T.-K. Lee, Invariance of minimal prime ideals under derivations, Proc. Amer. Math. Soc. 113 (1991), 613-616. Zbl0733.16012
  10. [10] J. Cusack, Automatic continuity and topologically simple radical Banach algebras, J. London Math. Soc. 16 (1977), 493-500. Zbl0398.46042
  11. [11] H. G. Dales, Automatic continuity: a survey, Bull. London Math. Soc. 10 (1978), 129-183. Zbl0391.46037
  12. [12] J. Dixmier, Algèbres envellopantes, Cahier Sci. 27, Gauthier-Villars, Paris, 1974. 
  13. [13] C. K. Fong and A. R. Sourour, On the operator identity A k X B k 0 , Canad. J. Math. 31 (1979), 845-857. Zbl0368.47024
  14. [14] R. V. Garimella, On nilpotency of the separating ideal of a derivation, Proc. Amer. Math. Soc. 117 (1993), 167-174. Zbl0794.46042
  15. [15] K. R. Goodearl and R. B. Warfield, Primitivity in differential operator rings, Math. Z. 180 (1982), 503-524. Zbl0495.16002
  16. [16] P. R. Halmos, Commutators of operators, II, Amer. J. Math. 76 (1954), 191-198. Zbl0055.10705
  17. [17] N. Jacobson, Rational methods in the theory of Lie algebras, Ann. of Math. 36 (1935), 875-881. Zbl0012.33704
  18. [18] B. E. Johnson, Continuity of derivations on commutative Banach algebras, Amer. J. Math. 91 (1969), 1-10. Zbl0181.41103
  19. [19] B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, ibid. 90 (1968), 1067-1073. Zbl0179.18103
  20. [20] I. Kaplansky, Functional analysis, in: Some Aspects of Analysis and Probability, Surveys Appl. Math. 4, New York, 1958, 1-34. 
  21. [21] D. C. Kleinecke, On operator commutators, Proc. Amer. Math. Soc. 8 (1957), 535-536. Zbl0079.12904
  22. [22] M. Mathieu, Is there an unbounded Kleinecke-Shirokov theorem?, Sem.ber. Funkt.anal. 18, Tübingen, 1990, 137-143. 
  23. [23] M. Mathieu, On the range of centralising derivations, preprint, 1991. 
  24. [24] M. Mathieu, Posner's second theorem deduced from the first, Proc. Amer. Math. Soc. 114 (1992), 601-602. Zbl0746.16030
  25. [25] M. Mathieu and G. J. Murphy, Derivations mapping into the radical, Arch. Math. (Basel) 57 (1991), 469-474. Zbl0714.46038
  26. [26] M. Mathieu and V. Runde, Derivations mapping into the radical, II, Bull. London Math. Soc. 24 (1992), 485-487. Zbl0733.46023
  27. [27] G. J. Murphy, Aspects of the theory of derivations, this volume, 267-275. Zbl0811.46045
  28. [28] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100. Zbl0082.03003
  29. [29] V. Pták, Commutators in Banach algebras, Proc. Edinburgh Math. Soc. 22 (1979), 207-211. Zbl0407.46043
  30. [30] C. R. Putnam, On the spectra of commutators, Proc. Amer. Math. Soc. 5 (1954), 929-931. Zbl0056.34301
  31. [31] V. Runde, Automatic continuity of derivations and epimorphisms, Pacific J. Math. 147 (1991), 365-374. Zbl0666.46052
  32. [32] V. Runde, Problems in automatic continuity, Ph.D. Thesis, Univ. California, Berkeley, 1993. 
  33. [33] V. Runde, Range inclusion results for derivations on noncommutative Banach algebras, Studia Math. 105 (1993), 159-172. Zbl0810.46044
  34. [34] G. Shilov, On a property of rings of functions, Dokl. Akad. Nauk SSSR 58 (1947), 985-988 (in Russian). 
  35. [35] F. V. Shirokov, Proof of a conjecture of Kaplansky, Uspekhi Mat. Nauk. 11 (1956), 167-168 (in Russian). Zbl0070.34201
  36. [36] A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20 (1969), 166-170. 
  37. [37] I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955), 260-264. Zbl0067.35101
  38. [38] J. G. Stampfli, On the range of a hyponormal derivation, Proc. Amer. Math. Soc. 52 (1975), 117-120. Zbl0315.47019
  39. [39] M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math. 128 (1988), 435-460. Zbl0681.47016
  40. [40] M. P. Thomas, Primitive ideals and derivations on non-commutative Banach algebras, Pacific J. Math. 159 (1993), 139-152. Zbl0739.47014
  41. [41] Yu. V. Turovskiĭ and V. S. Shul'man, Conditions for massiveness of the range of the derivation of a Banach algebra and associated differential operators, Math. Notes 42 (1987), 669-674. 
  42. [42] I. Vidav, Über eine Vermutung von Kaplansky, Math. Z. 62 (1955), 330. 
  43. [43] J. Vukman, Commuting and centralizing mappings in prime rings, Proc. Amer. Math. Soc. 109 (1990), 47-52. Zbl0697.16035
  44. [44] J. Vukman, On derivations in prime rings and Banach algebras, ibid. 116 (1992), 877-884. Zbl0792.16034
  45. [45] J. Vukman, A result concerning derivations in noncommutative Banach algebras, Glas. Mat. 26 (1991), 83-88. Zbl0813.46037
  46. [46] H. Wielandt, Über die Unbeschränktheit der Operatoren der Quantenmechanik, Math. Ann. 121 (1949/50), 21. Zbl0035.19903
  47. [47] J. P. Williams, On the range of a derivation, Pacific J. Math. 38 (1971), 273-279. Zbl0205.42102
  48. [48] A. Wintner, The unboundedness of quantum-mechanical matrices, Phys. Rev. 71 (1947), 738-739. Zbl0032.13602
  49. [49] B. Yood, Continuous homomorphisms and derivations on Banach algebras, in: F. Greenleaf and D. Gulick (eds.), Banach Algebras and Several Complex Variables, Contemp. Math. 32, Amer. Math. Soc., Providence, R.I., 1984, 279-284. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.