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Skew Derivations in Banach Algebras

Published online by Cambridge University Press:  10 June 2015

Pao-Kuei Liau  and
Cheng-Kai Liu
Pao-Kuei Liau
Affiliation:
Department of Mathematics, National Changhua University of Education, Changhua 500, Taiwan ([email protected])
Cheng-Kai Liu
Affiliation:
Department of Mathematics, National Changhua University of Education, Changhua 500, Taiwan ([email protected])
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Abstract

We investigate the global versions of the Kleinecke–Shirokov theorem for skew derivations in Banach algebras. Centralizing skew derivations on Banach algebras are also studied.

Keywords

Banach algebraquasinilpotentradicalskew derivationPrimary 47B4747B4846H15
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 3 , October 2015 , pp. 683 - 696
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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References

1.Abdeliali, Z., On Φ-derivations in Banach algebras, Commun. Alg. 34 (2006), 2437–2452.CrossRefGoogle Scholar
2.Beidar, K. I. and Brešar, M., Extended Jacobson density theorem for rings with automorphisms and derivations, Israel J. Math. 122 (2001), 317–346.CrossRefGoogle Scholar
3.Beidar, K. I., Fong, Y., Ke, W.-F. and Lee, C.-H., Posner’s theorem for generalized (σ, τ)-derivations, in Lie algebras, rings and related topics (ed. Fong, Y., Mikhalev, A. A. and Zelmanov, E.), pp. 5–12 (Springer, 2000).Google Scholar
4.Brešar, M., Centralizing mappings on von Neumann algebras, Proc. Am. Math. Soc. 111 (1991), 501–510.Google Scholar
5.Brešar, M., Derivations of noncommutative Banach algebras, II, Arch. Math. 63 (1994), 56–59.Google Scholar
6.Brešar, M., On automorphisms of Banach algebras, Arch. Math. 78 (2002), 297–302.Google Scholar
7.Brešar, M. and Šemrl, P., Spectral characterizations of central elements in Banach algebras, Studia Math. 120 (1996), 47–52.CrossRefGoogle Scholar
8.Brešar, M. and Šemrl, P., On locally linearly dependent operators and derivations, Trans. Am. Math. Soc. 351 (1999), 1257–1275.Google Scholar
9.Brešar, M. and Villena, A. R., The noncommutative Singer–Wermer conjecture and φ-derivations, J. Lond. Math. Soc. 66 (2002), 710–720.Google Scholar
10.Brešar, M. and Vukman, J., On left derivations and related mappings, Proc. Am. Math. Soc. 110 (1990), 7–16.CrossRefGoogle Scholar
11.Brešar, M. and Vukman, J., Derivations of noncommutative Banach algebras, Arch. Math. 59 (1992), 363–370.Google Scholar
12.Brešar, M., Fošner, A. and Fošner, M., A Kleinecke–Shirokov type condition with Jordan automorphism, Studia Math. 147 (2001), 237–242.Google Scholar
13.Chebotar, M. A., Ke, W.-F. and Lee, P.-H., On a Brešar–Šemrl conjecture and derivations of Banach algebras, Q. J. Math. 57 (2006), 1–10.Google Scholar
14.Chuang, C.-L. and Liu, C.-K., Extended Jacobson density theorem for rings with skew derivations, Commun. Alg. 35 (2007), 1391–1413.CrossRefGoogle Scholar
15.Cusak, J., Automatic continuity and topological simple radical Banach algebras, J. Lond. Math. Soc. 16 (1977), 493–500.Google Scholar
16.Fack, T., Finite sums of commutators in C *-algebras, Annales Inst. Fourier 32 (1982), 129–137.Google Scholar
17.Goodearl, K. R. and Letzter, E. S., Prime ideals in skew and q-skew polynomial rings, Memoirs of the American Mathematical Society, Volume 109 (American Mathematical Society, Providence, RI, 1994).Google Scholar
18.Hejazian, S. and Janfada, A. R., Invariance of primitive ideals by φ-derivations on Banach algebras, Taiwan. J. Math. 13 (2009), 1181–1187.Google Scholar
19.Kleinecke, D. C., On operator commutators, Proc. Am. Math. Soc. 8 (1957), 535–536.Google Scholar
20.Lee, T.-K., Derivations on noncommutative Banach algebras, Studia Math. 167 (2005), 153–160.Google Scholar
21.Lee, T.-K. and Liu, C.-K., Spectrally bounded φ-derivations on Banach algebras, Proc. Am. Math. Soc. 133 (2005), 1427–1435.CrossRefGoogle Scholar
22.Lee, T.-K. and Liu, C.-K., Partially defined σ-derivations on Banach algebras, Studia Math. 190 (2009), 661–669.Google Scholar
23.Lee, P.-H. and Liu, C.-K., On the composition of q-skew derivations in Banach algebras, Linear Alg. Applic. 434 (2011), 2413–2429.Google Scholar
24.Liu, C.-K., Skew derivations with nilpotent values in rings and Banach algebras, Commun. Alg. 40 (2012), 4336–4345.Google Scholar
25.Mathieu, M., Where to find the image of a derivation, in Functional analysis and operator theory, Banach Center Publications, Volume 30, pp. 237–249 (Polish Academy of Science, Warsaw, 1994).Google Scholar
26.Mathieu, M. and Murphy, G. J., Derivations mapping into the radical, Arch. Math. 57 (1991), 469–474.Google Scholar
27.Mathieu, M. and Runde, V., Derivations mapping into the radical, II, Bull. Lond. Math. Soc. 24 (1992), 485–487.Google Scholar
28.Mirzavaziri, M. and Moslehian, M. S., Automatic continuity of σ-derivations on C *-algebra, Proc. Am. Math. Soc. 134 (2006), 3319–3327.Google Scholar
29.Pop, C., Finite sums of commutators, Proc. Am. Math. Soc. 130 (2002), 3039–3041.CrossRefGoogle Scholar
30.Pták, V., Derivations, commutators and the radical, Manuscr. Math. 23 (1978), 355–362.Google Scholar
31.Pták, V., Commutators in Banach algebras, Proc. Edinb. Math. Soc. 22 (1979), 207–211.Google Scholar
32.Richart, C. E., General theory of Banach algebras (Van Nostrand, New York, 1960).Google Scholar
33.Shirokov, F. V., Proof of a conjecture of Kaplansky, Usp. Mat. Nauk 11 (1956), 167–168.Google Scholar
34.Sinclair, A. M., Continuous derivations on Banach algebras, Proc. Am. Math. Soc. 20 (1969), 166–170.Google Scholar
35.Sinclair, A. M., Automatic continuity of linear operators, London Mathematical Society Lecture Note Series, Volume 21 (Cambridge University Press, 1976).Google Scholar
36.Singer, I. M. and Wermer, J., Derivations on commutative normed algebras, Math. Annalen 129 (1955), 260–264.CrossRefGoogle Scholar
37.Thomas, M. P., The image of a derivation is contained in the radical, Annals Math. 128 (1988), 435–460.Google Scholar
38.Thomas, M. P., Primitive ideals and derivations on noncommutative Banach algebras, Pac. J. Math. 159 (1993), 139–152.Google Scholar
39.Turovskii, Yu.V. and Shul’man, V. S., Conditions for the massiveness of the range of a derivation of a Banach algebra and of associated differential operators, Mat. Zametki 42 (1987), 305–314.Google Scholar