Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-15T01:36:01.114Z Has data issue: false hasContentIssue false
  • English
  • Français

On the norm of an inner derivation of a C*-algebra

Published online by Cambridge University Press:  24 October 2008

R. J. Archbold
R. J. Archbold
Affiliation:
The University of Aberdeen
Get access Rights & Permissions [Opens in a new window]

Extract

Let A be a C*-algebra with centre Z. If a ∈ A, the bounded linear mapping x → + ax – xa (x ∈ A) is called the inner derivation of A induced by a, and we denote it by D(a, A). A simple application of the triangle inequality shows that

where d (a; Z) denotes the distance from a to Z in the normed space A.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 84 , Issue 2 , September 1978 , pp. 273 - 291
Copyright
Copyright © Cambridge Philosophical Society 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Archbold, R. J. Certain properties of operator algebras, Ph.D. Thesis (University of Newcastle upon Tyne, 1972).Google Scholar
(2)Archbold, R. J.On the centre of a tensor product of C*-algebras. J. London Math. Soc. (2) 10 (1975), 257–262.CrossRefGoogle Scholar
(3)Archbold, R. J.An averaging process for C*-algebras related to weighted shifts. Proc. London Math. Soc. (in the Press).Google Scholar
(4)Batty, C. J. K. Derivations of tensor products of C*-algebras (in the Press).Google Scholar
(5)Bratelli, O.Inductive limits of finite dimensionai C*-algebras. Trans. Amer. Math. Soc. 171 (1972), 195–234.Google Scholar
(6)Bratelli, O.The center of approximately finite-dimensional C*-algebras. J. Functional Analysis 21 (1976), 195–202.CrossRefGoogle Scholar
(7)Dauns, J. and Hofmann, K. H.Representations of rings by continuous sections. Mem. Amer. Math. Soc. 83 (1968).Google Scholar
(8)Delaroche, C.Sur les centres dea C*-algèbres. Bull. Sci. Math. 91 (1967), 105–112.Google Scholar
(9)Dixmier, J.Lea C*-algèbresetleurs représentation, 2me édition (Paris: Gauthier–Villars, 1969).Google Scholar
(10)Dixmier, J.Lea algèbres d'opérateurs dans l'espace Hilbertien, 2me édition (Paris: Gauthier-Villars, 1969).Google Scholar
(11)Dixmier, J.Ideal centre of a C*-algebra. Duke Math. J. 35 (1968), 375–382.CrossRefGoogle Scholar
(12)Elliott, G.A. Some C*-algebras with outerderivations. Rocky Mountain J. Math. 3 (1973), 501–506.CrossRefGoogle Scholar
(13)Fell, J. M. G.The structure of algebras of operator fields. Acta Math. 106 (1961), 233–280.CrossRefGoogle Scholar
(14)Gajendragadkar, P. Thesis (Indiana University, Bloomington, Ind., 1970).Google Scholar
(15)Green, W. L.Topological dynamics and C*-algebras. Trans. Amer. Math. Soc. 210 (1975), 107–121.Google Scholar
(16)Hall, A. A. Ph.D. Thesis (University of Newcastle upon Tyne, 1972).Google Scholar
(17)Hall, A. A.Derivations of certain C*-algebras. J. London Math. Soc. (2) 5 (1972), 321–329.CrossRefGoogle Scholar
(18)Halpern, H. The norm of an inner derivation of an AW*-algebra (in the Press).Google Scholar
(19)Jameson, G. J. O.Topology and normed spaces (London: Chapman and Hall, 1974).Google Scholar
(20)Kadison, R. V., Lance, E. C. and Ringrose, J. R.Derivations and automorphisms of operator algebras II. J. Functional Analysis 1 (1967), 204–221.CrossRefGoogle Scholar
(21)Kaplansky, I.Normed algebras. Duke Math. J. 16 (1949), 399–418.CrossRefGoogle Scholar
(22)Kaplansky, I.The structure of certain operator algebras. Trans. Amer. Math. Soc. 70 (1951), 219–255.CrossRefGoogle Scholar
(23)Misonou, Y.On a weakly central operator algebra. Tóhoku Math. J. 4 (1952), 194–202.Google Scholar
(24)Ringrose, J. R. Derivations of quotients of von Neumann algebras (in the Press).Google Scholar
(25)Sakai, S.On a conjecture of Kaplansky. Tóhoku Math. J. 12 (1960), 31–33.Google Scholar
(26)Sproston, J. P.Derivations and automorphisms of homogeneous C*-algebras. Proc. London Math. Soc. (3) 32 (1976), 521–536.CrossRefGoogle Scholar
(27)Stampfli, J. G.The norm of a derivation. Pacific J. Math. 33 (1970), 737–747.CrossRefGoogle Scholar
(28)Vesterstrøm, J.On the homomorphic image of the centre of a C*-algebra. Math. Scand. 29 (1971), 134–136.CrossRefGoogle Scholar
(29)Zsido, L.The norm of a derivation in a W*-algebra, Proc. Amer. Math. Soc. 38 (1973), 147–150.Google Scholar