Where to find the image of a derivation
Mathieu, Martin
Banach Center Publications, Tome 29 (1994), p. 237-249 / Harvested from The Polish Digital Mathematics Library

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.

Publié le : 1994-01-01
EUDML-ID : urn:eudml:doc:262681
@article{bwmeta1.element.bwnjournal-article-bcpv30z1p237bwm,
     author = {Mathieu, Martin},
     title = {Where to find the image of a derivation},
     journal = {Banach Center Publications},
     volume = {29},
     year = {1994},
     pages = {237-249},
     zbl = {0813.47043},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv30z1p237bwm}
}
Mathieu, Martin. Where to find the image of a derivation. Banach Center Publications, Tome 29 (1994) pp. 237-249. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv30z1p237bwm/

[000] [1] C. Apostol, Inner derivations with closed range, Rev. Roumaine Math. Pures Appl. 21 (1976), 242-265.

[001] [2] C. Apostol and J. G. Stampfli, On derivation ranges, Indiana Univ. Math. J. 25 (1976), 857-869. | Zbl 0355.47025

[002] [3] J. Bergen, I. N. Herstein and C. Lanski, Derivations with invertible values, Canad. J. Math. 35 (1983), 300-310. | Zbl 0522.16031

[003] [4] M. Brešar, Centralizing mappings on von Neumann algebras, Proc. Amer. Math. Soc. 111 (1991), 501-510. | Zbl 0746.46054

[004] [5] M. Brešar, On a generalization of the notion of centralizing mappings, ibid. 114 (1992), 641-649. | Zbl 0754.16020

[005] [6] M. Brešar, Derivations decreasing the spectral radius, Arch. Math. (Basel) 61 (1993), 160-162. | Zbl 0818.46049

[006] [7] M. Brešar and J. Vukman, On left derivations and related mappings, Proc. Amer. Math. Soc. 110 (1990), 7-16. | Zbl 0703.16020

[007] [8] M. Brešar and J. Vukman, Derivations on noncommutative Banach algebras, Arch. Math. (Basel) 59 (1992), 363-370. | Zbl 0807.46049

[008] [9] C.-L. Chuang and T.-K. Lee, Invariance of minimal prime ideals under derivations, Proc. Amer. Math. Soc. 113 (1991), 613-616. | Zbl 0733.16012

[009] [10] J. Cusack, Automatic continuity and topologically simple radical Banach algebras, J. London Math. Soc. 16 (1977), 493-500. | Zbl 0398.46042

[010] [11] H. G. Dales, Automatic continuity: a survey, Bull. London Math. Soc. 10 (1978), 129-183. | Zbl 0391.46037

[011] [12] J. Dixmier, Algèbres envellopantes, Cahier Sci. 27, Gauthier-Villars, Paris, 1974.

[012] [13] C. K. Fong and A. R. Sourour, On the operator identity AkXBk0, Canad. J. Math. 31 (1979), 845-857. | Zbl 0368.47024

[013] [14] R. V. Garimella, On nilpotency of the separating ideal of a derivation, Proc. Amer. Math. Soc. 117 (1993), 167-174. | Zbl 0794.46042

[014] [15] K. R. Goodearl and R. B. Warfield, Primitivity in differential operator rings, Math. Z. 180 (1982), 503-524. | Zbl 0495.16002

[015] [16] P. R. Halmos, Commutators of operators, II, Amer. J. Math. 76 (1954), 191-198. | Zbl 0055.10705

[016] [17] N. Jacobson, Rational methods in the theory of Lie algebras, Ann. of Math. 36 (1935), 875-881. | Zbl 0012.33704

[017] [18] B. E. Johnson, Continuity of derivations on commutative Banach algebras, Amer. J. Math. 91 (1969), 1-10. | Zbl 0181.41103

[018] [19] B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, ibid. 90 (1968), 1067-1073. | Zbl 0179.18103

[019] [20] I. Kaplansky, Functional analysis, in: Some Aspects of Analysis and Probability, Surveys Appl. Math. 4, New York, 1958, 1-34.

[020] [21] D. C. Kleinecke, On operator commutators, Proc. Amer. Math. Soc. 8 (1957), 535-536. | Zbl 0079.12904

[021] [22] M. Mathieu, Is there an unbounded Kleinecke-Shirokov theorem?, Sem.ber. Funkt.anal. 18, Tübingen, 1990, 137-143.

[022] [23] M. Mathieu, On the range of centralising derivations, preprint, 1991.

[023] [24] M. Mathieu, Posner's second theorem deduced from the first, Proc. Amer. Math. Soc. 114 (1992), 601-602. | Zbl 0746.16030

[024] [25] M. Mathieu and G. J. Murphy, Derivations mapping into the radical, Arch. Math. (Basel) 57 (1991), 469-474. | Zbl 0714.46038

[025] [26] M. Mathieu and V. Runde, Derivations mapping into the radical, II, Bull. London Math. Soc. 24 (1992), 485-487. | Zbl 0733.46023

[026] [27] G. J. Murphy, Aspects of the theory of derivations, this volume, 267-275. | Zbl 0811.46045

[027] [28] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100. | Zbl 0082.03003

[028] [29] V. Pták, Commutators in Banach algebras, Proc. Edinburgh Math. Soc. 22 (1979), 207-211. | Zbl 0407.46043

[029] [30] C. R. Putnam, On the spectra of commutators, Proc. Amer. Math. Soc. 5 (1954), 929-931. | Zbl 0056.34301

[030] [31] V. Runde, Automatic continuity of derivations and epimorphisms, Pacific J. Math. 147 (1991), 365-374. | Zbl 0666.46052

[031] [32] V. Runde, Problems in automatic continuity, Ph.D. Thesis, Univ. California, Berkeley, 1993.

[032] [33] V. Runde, Range inclusion results for derivations on noncommutative Banach algebras, Studia Math. 105 (1993), 159-172. | Zbl 0810.46044

[033] [34] G. Shilov, On a property of rings of functions, Dokl. Akad. Nauk SSSR 58 (1947), 985-988 (in Russian).

[034] [35] F. V. Shirokov, Proof of a conjecture of Kaplansky, Uspekhi Mat. Nauk. 11 (1956), 167-168 (in Russian). | Zbl 0070.34201

[035] [36] A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20 (1969), 166-170.

[036] [37] I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955), 260-264. | Zbl 0067.35101

[037] [38] J. G. Stampfli, On the range of a hyponormal derivation, Proc. Amer. Math. Soc. 52 (1975), 117-120. | Zbl 0315.47019

[038] [39] M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math. 128 (1988), 435-460. | Zbl 0681.47016

[039] [40] M. P. Thomas, Primitive ideals and derivations on non-commutative Banach algebras, Pacific J. Math. 159 (1993), 139-152. | Zbl 0739.47014

[040] [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.

[041] [42] I. Vidav, Über eine Vermutung von Kaplansky, Math. Z. 62 (1955), 330.

[042] [43] J. Vukman, Commuting and centralizing mappings in prime rings, Proc. Amer. Math. Soc. 109 (1990), 47-52. | Zbl 0697.16035

[043] [44] J. Vukman, On derivations in prime rings and Banach algebras, ibid. 116 (1992), 877-884. | Zbl 0792.16034

[044] [45] J. Vukman, A result concerning derivations in noncommutative Banach algebras, Glas. Mat. 26 (1991), 83-88. | Zbl 0813.46037

[045] [46] H. Wielandt, Über die Unbeschränktheit der Operatoren der Quantenmechanik, Math. Ann. 121 (1949/50), 21. | Zbl 0035.19903

[046] [47] J. P. Williams, On the range of a derivation, Pacific J. Math. 38 (1971), 273-279. | Zbl 0205.42102

[047] [48] A. Wintner, The unboundedness of quantum-mechanical matrices, Phys. Rev. 71 (1947), 738-739. | Zbl 0032.13602

[048] [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.