Hostname: page-component-84b7d79bbc-g5fl4 Total loading time: 0 Render date: 2024-07-31T02:31:38.435Z Has data issue: false hasContentIssue false
  • English
  • Français

Derivations, local derivations and atomic boolean subspace lattices

Published online by Cambridge University Press:  17 April 2009

Pengtong Li  and
Jipu Ma
Pengtong Li
Affiliation:
Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, People's Republic of China e-mail: ptli@nju.edu.cn, pengtonglee@sina.com.cn
Jipu Ma
Affiliation:
Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, People's Republic of China
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.

Let ℒ be an atomic Boolean subspace lattice on a Banach space X. In this paper, we prove that if ℳ is an ideal of Alg ℒ then every derivation δ from Alg ℒ into ℳ is necessarily quasi-spatial, that is, there exists a densely defined closed linear operator T: 𝒟(T) ⊆ X → X with its domain 𝒟(T) invariant under every element of Alg ℒ, such that δ(A) x = (TA – AT) x for every A ∈ Alg ℒ and every x ∈ 𝒟(T). Also, if ℳ ⊆ ℬ(X) is an Alg ℒ-module then it is shown that every local derivation from Alg ℒ into ℳ is necessary a derivation. In particular, every local derivation from Alg ℒ into ℬ(X) is a derivation and every local derivation from Alg ℒ into itself is a quasi-spatial derivation.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 66 , Issue 3 , December 2002 , pp. 477 - 486
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Argyros, S., Lambrou, M.S. and Longstaff, W.E., ‘Atomic Boolean subspace lattices and applications to the theory of bases’, Mem. Amer. Math. Soc. 91 (1991).Google Scholar
[2]Han, D. and Wei, S., ‘Local derivations of nest algebras’, Proc. Amer. Math. Soc. 123 (1995), 3095–3100.CrossRefGoogle Scholar
[3]Jing, W., ‘Local derivations of reflexive algebras’, Proc. Amer. Math. Soc. 125 (1997), 869–873.Google Scholar
[4]Jing, W., ‘Local derivations of reflexive algebras II’, Proc. Amer. Math. Soc. 129 (2001), 1733–1737.Google Scholar
[5]Johnson, B.E., ‘Local derivations on C*–algebras axe derivations’, Trans. Amer. Math. Soc. 353 (2001), 313–325.CrossRefGoogle Scholar
[6]Kadison, R.V., ‘Local derivations’, J. Algebra 130 (1990), 494–509.CrossRefGoogle Scholar
[7]Katavolos, A., Lambrou, M.S. and Longstaff, W.E., ‘Pentagon subspace lattices on Banach spaces’, J. Operator Theory 46 (2001), 355–380.Google Scholar
[8]Lambrou, M.S., ‘Approximants, commutants and double commutants in normed algebras’, J. London Math. Soc. 2 25 (1982), 499–512.CrossRefGoogle Scholar
[9]Lambrou, M.S., On some reflexive lattices and related algebras, Ph. D. dissertation (University of London, Kings College, 1977).Google Scholar
[10]Lambrou, M.S., ‘On the rank of operators in reflexive algebras’ Linear Algebra Appl. 142 (1990), 211–235.CrossRefGoogle Scholar
[11]Larson, D.R. and Sourour, A.R., ‘Local derivations and local automorphisms of B (X)’ Proc. Sympos. Pure Math. 51 (1990), 187–194.CrossRefGoogle Scholar
[12]Longstaff, W.E., ‘Strongly reflexive lattices’, J. London Math. Soc. (2) 11 (1975), 491–498.CrossRefGoogle Scholar
[13]Panaia, O., ‘Algebraic isomorphisms and finite distributive subspace lattices’, J. London Math. Soc. (2) 59 (1999), 1033–1048.CrossRefGoogle Scholar