Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-24T18:20:24.799Z Has data issue: false hasContentIssue false
  • English
  • Français

On the distance of the composition of two derivations to the generalized derivations

Published online by Cambridge University Press:  18 May 2009

Matej Bresar
Matej Bresar
Affiliation:
Institute of Mathematics, Physics and Mechanics, University of Ljubljana, P.O. Box 543, 61111 Ljubljana, Yugoslavia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A well-known theorem of E. Posner [10] states that if the composition d1d2 of derivations d1d2 of a prime ring A of characteristic not 2 is a derivation, then either d1 = 0 or d2 = 0. A number of authors have generalized this theorem in several ways (see e.g. [1], [2], and [5], where further references can be found). Under stronger assumptions when A is the algebra of all bounded linear operators on a Banach space (resp. Hilbert space), Posner's theorem was reproved in [3] (resp. [12]). Recently, M. Mathieu [8] extended Posner's theorem to arbitrary C*-algebras.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 33 , Issue 1 , January 1991 , pp. 89 - 93
Copyright
Copyright © Glasgow Mathematical Journal Trust 1991

References

REFERENCES

1.Bergen, J., Herstein, I. N. and Kerr, J., Lie ideals and derivations of prime rings, J. Algebra 71 (1981), 259–267.CrossRefGoogle Scholar
2.Brěsar, M. and Vukman, J., Orthogonal derivations and an extension of a theorem of Posner, to appear in Rad. Mat.Google Scholar
3.Fong, C. K. and Sourour, A. R., On the operator identity ΣAkXBk ≡ = 0, Canad. J. Math. 31 (1979), 845–857.CrossRefGoogle Scholar
4.Gajendragadkar, P., Norm of a derivation on a von Neumann algebra, Trans. Amer. Math. Soc. 170 (1972), 165–170.CrossRefGoogle Scholar
5.Lanski, C., Differential identities, Lie ideals, and Posner's theorems, Pacific J. Math. 134 (1988), 275–297.Google Scholar
6.Mathieu, M., Applications of ultraprime Banach algebras in the theory of elementary operators, Thesis (Tübingen, 1986).Google Scholar
7.Mathieu, M., Elementary operators on prime C*-algebras I, Math. Ann. 284 (1989), 223–244.CrossRefGoogle Scholar
8.Mathieu, M., Properties of the product of two derivations of a C*-algebra, to appear in Canad. Math. Bull.Google Scholar
9.Mathieu, M., Rings of quotients of ultraprime Banach algebras. With applications to elementary operators, to appear in the Proceedings of the Conference on Banach algebras and Automatic Continuity (Canberra, Australia, 1989).Google Scholar
10.Posner, E., Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093–1100.CrossRefGoogle Scholar
11.Sakai, S., C*-algebras and W*-algebras (Springer-Verlag, 1971).Google Scholar
12.Williams, J. P., On the range of a derivation, Pacific J. Math. 38 (1971), 273–279.CrossRefGoogle Scholar
13.Zsidó, L., The norm of a derivation in a W*-algebra, Proc. Amer. Math. Soc. 38 (1973), 147–150.Google Scholar