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Continuity of Lie derivations on Banach algebras. (English) Zbl 0957.46035
A linear map $$D$$ from a Banach algebra $$A$$ to itself which satisfies $$D([a,b])=[D(a),b]+[a,D(b)]$$, where $$[a,b]:=ab-ba$$ is the usual Lie bracket, is called a Lie derivation. Denote by $${\mathcal S}(D)$$ the separating space of the map $$D$$ and by $${\mathcal Z}(A)$$ the centre of $$A$$. The authors prove the following
Theorem. Let $$D$$ be a derivation on a semisimple Banach algebra $$A$$. Then $${\mathcal S}(D)\subset {\mathcal Z}(A)$$.
They also give an example of a discontinuous derivation on a semisimple Banach algebra whose centre is $$\mathbb C$$. For an extension of these results, see also B. Aupetit and M. Mathieu, Stud. Math. 138, No. 2, 193-199 (2000; Zbl 0962.46038).

##### MSC:
 46H40 Automatic continuity 16W10 Rings with involution; Lie, Jordan and other nonassociative structures 46H70 Nonassociative topological algebras
##### Keywords:
Lie derivation; Lie product; automatic continuity
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##### References:
 [1] DOI: 10.2307/2373290 · Zbl 0179.18103 · doi:10.2307/2373290 [2] DOI: 10.1090/S0002-9904-1961-10666-6 · Zbl 0107.02704 · doi:10.1090/S0002-9904-1961-10666-6 [3] de la Harpe, Classical Banach-Lie algebras and Banach Lie groups of operators in Hilbert space 285 (1972) · Zbl 0256.22015 · doi:10.1007/BFb0071306 [4] DOI: 10.2307/2154392 · Zbl 0791.16028 · doi:10.2307/2154392 [5] DOI: 10.1307/mmj/1028999091 · Zbl 0123.03201 · doi:10.1307/mmj/1028999091 [6] Thomas, Pacific J. Math. 159 pp 139– (1993) · Zbl 0739.47014 · doi:10.2140/pjm.1993.159.139 [7] DOI: 10.2307/2042216 · Zbl 0384.46047 · doi:10.2307/2042216 [8] DOI: 10.1215/S0012-7094-73-04032-5 · Zbl 0264.46064 · doi:10.1215/S0012-7094-73-04032-5 [9] DOI: 10.2307/1995366 · Zbl 0192.37802 · doi:10.2307/1995366 [10] Palmer, Banach Algebras and the General Theory of *-algebras. Volume I: Algebras and Banach Algebras (1994) · Zbl 0809.46052 · doi:10.1017/CBO9781107325777
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