Lie derivations on Banach algebras. (English) Zbl 0957.46032

Villena, A. R.

J. Algebra 226, No. 1, 390-409 (2000).

A Lie derivation \(\Delta\) on a unital complex Banach algebra \(A\) is considered. The main result of the article is the following
Theorem. Let \(D\) be a Lie derivation on a unital complex Banach algebra \(A\). Then for every primitive ideal \(P\) of \(A\), except for a finite set of them which have finite codimension greater than one, there exist a derivation \(d\) from \(A/P\) to itself and a linear functional \(\tau\) on \(A\) such that \[ Q_p\Delta(a)= d(a+ P)+ \tau(a) \] for all \(a\in A\) (where \(Q_p\) denotes the quotient map from \(A\) onto \(A(p)\).
Moreover, the preceding decomposition holds for all primitive ideals in the case where \(\Delta\) is continuous.

Reviewer: Andreiy Kondrat’yev (Pensacola)

MSC:

46H05

General theory of topological algebras

Keywords:

Lie derivation; unital complex Banach algebra; primitive ideal

References:

[1]	Bade, W.G.; Curtis, P.C., Prime ideals and automatic continuity problems for Banach algebras, J. funct. anal., 29, 88-103, (1978) · Zbl 0399.46036
[2]	Banning, R.; Mathieu, M., Commutativity preserving mappings on semiprime rings, Comm. algebra, 25, 247-265, (1997) · Zbl 0865.16015
[3]	Berenguer, M.I.; Villena, A.R., Continuity of Lie derivations on Banach algebras, Proc. Edinburgh math. soc., 41, 625-630, (1998) · Zbl 0957.46035
[4]	Brešar, M., Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. amer. math. soc., 335, 525-546, (1993) · Zbl 0791.16028
[5]	Carpenter, R.L., Continuity of systems of derivations on F-algebras, Proc. amer. math. soc., 30, 141-146, (1971) · Zbl 0221.46044
[6]	Herstein, I.N., Lie and Jordan structures in simple, associative rings, Bull. amer. math. soc., 67, 517-531, (1961) · Zbl 0107.02704
[7]	Jacobson, N., Lie algebras, (1962), Dover New York · JFM 61.1044.02
[8]	Johnson, B.E.; Sinclair, A.M., Continuity of derivations and a problem of Kaplansky, Amer. J. math., 90, 1067-1073, (1968) · Zbl 0179.18103
[9]	Loy, R.J., Continuity of higher derivations, Proc. amer. math. soc., 37, 505-510, (1973) · Zbl 0252.46045
[10]	Mathieu, M., Where to find the image of a derivation, Banach center publications, 30, (1994), PWN Warsaw, p. 237-249 · Zbl 0813.47043
[11]	Martindale, W.S., Lie derivations of primitive rings, Michigan J. math., 11, 183-187, (1964) · Zbl 0123.03201
[12]	Miller, J.B., Higher derivations on Banach algebras, Amer. J. math., 92, 301-331, (1970) · Zbl 0201.17203
[13]	Palmer, T.W., Banach algebras and the general theory of ∗-algebras, Volume I: algebras and Banach algebras, I, (1994), Cambridge Univ. Press Cambridge · Zbl 0809.46052
[14]	Singer, I.M.; Wermer, J., Derivations on commutative normed algebras, Math. ann., 129, 260-264, (1955) · Zbl 0067.35101
[15]	Thomas, M.P., The image of a derivation is contained in the radical, Ann. math., 128, 435-460, (1988) · Zbl 0681.47016
[16]	Thomas, M.P., Primitive ideals and derivations on non-commutative Banach algebras, Pacific J. math., 159, 139-152, (1993) · Zbl 0739.47014