Lie derivations of 𝒥-subspace lattice algebras

Lu, Fangyan

doi:10.1090/S0002-9939-07-08767-9

Lie derivations of $\mathcal J$-subspace lattice algebras
HTML articles powered by AMS MathViewer

by Fangyan Lu PDF

Proc. Amer. Math. Soc. 135 (2007), 2581-2590 Request permission

Abstract:

We describe the structure of Lie derivations of $\mathcal J$-subspace lattice algebras. The results can apply to atomic Boolean subspace lattice algebras and pentagon subspace lattice algebras, respectively.

References

J. Alaminos, M. Mathieu, and A. R. Villena, Symmetric amenability and Lie derivations, Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 2, 433–439. MR 2092069, DOI 10.1017/S0305004104007637
K. I. Beidar and M. A. Chebotar, On Lie derivations of Lie ideals of prime algebras, Israel J. Math. 123 (2001), 131–148. MR 1835291, DOI 10.1007/BF02784122
Matej Brešar, Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. Amer. Math. Soc. 335 (1993), no. 2, 525–546. MR 1069746, DOI 10.1090/S0002-9947-1993-1069746-X
Matej Brešar and Peter emrl, Mappings which preserve idempotents, local automorphisms, and local derivations, Canad. J. Math. 45 (1993), no. 3, 483–496. MR 1222512, DOI 10.4153/CJM-1993-025-4
Wai-Shun Cheung, Lie derivations of triangular algebras, Linear Multilinear Algebra 51 (2003), no. 3, 299–310. MR 1995661, DOI 10.1080/0308108031000096993
Kenneth R. Davidson, Nest algebras, Pitman Research Notes in Mathematics Series, vol. 191, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. Triangular forms for operator algebras on Hilbert space. MR 972978
Don Hadwin and Jiankui Li, Local derivations and local automorphisms, J. Math. Anal. Appl. 290 (2004), no. 2, 702–714. MR 2033052, DOI 10.1016/j.jmaa.2003.10.015
Paul Richard Halmos, A Hilbert space problem book, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 17, Springer-Verlag, New York-Berlin, 1982. MR 675952
B. E. Johnson, Symmetric amenability and the nonexistence of Lie and Jordan derivations, Math. Proc. Cambridge Philos. Soc. 120 (1996), no. 3, 455–473. MR 1388200, DOI 10.1017/S0305004100075010
B. E. Johnson, Local derivations on $C^*$-algebras are derivations, Trans. Amer. Math. Soc. 353 (2001), no. 1, 313–325. MR 1783788, DOI 10.1090/S0002-9947-00-02688-X
Richard V. Kadison, Local derivations, J. Algebra 130 (1990), no. 2, 494–509. MR 1051316, DOI 10.1016/0021-8693(90)90095-6
A. Katavolos, M. S. Lambrou, and W. E. Longstaff, Pentagon subspace lattices on Banach spaces, J. Operator Theory 46 (2001), no. 2, 355–380. MR 1870412
M. S. Lambrou, Approximants, commutants and double commutants in normed algebras, J. London Math. Soc. (2) 25 (1982), no. 3, 499–512. MR 657507, DOI 10.1112/jlms/s2-25.3.499
M. S. Lambrou, Automatic continuity and implementation of homomorphisms, (manuscript).
M. S. Lambrou and W. E. Longstaff, Non-reflexive pentagon subspace lattices, Studia Math. 125 (1997), no. 2, 187–199. MR 1455633, DOI 10.4064/sm-125-2-187-199
W. E. Longstaff, Strongly reflexive lattices, J. London Math. Soc. (2) 11 (1975), no. 4, 491–498. MR 394233, DOI 10.1112/jlms/s2-11.4.491
W. E. Longstaff, J. B. Nation, and Oreste Panaia, Abstract reflexive sublattices and completely distributive collapsibility, Bull. Austral. Math. Soc. 58 (1998), no. 2, 245–260. MR 1642047, DOI 10.1017/S0004972700032226
W. E. Longstaff and Oreste Panaia, $\scr J$-subspace lattices and subspace $\rm M$-bases, Studia Math. 139 (2000), no. 3, 197–212. MR 1762581
Fangyan Lu and Pengtong Li, Algebraic isomorphisms and Jordan derivations of $\scr J$-subspace lattice algebras, Studia Math. 158 (2003), no. 3, 287–301. MR 2014303, DOI 10.4064/sm158-3-7
Fangyan Lu and Pengtong Li, Jordan isomorphisms of $\scr J$-subspace lattice algebras, Linear Algebra Appl. 371 (2003), 255–264. MR 1997374, DOI 10.1016/S0024-3795(03)00461-0
Wallace S. Martindale III, Lie derivations of primitive rings, Michigan Math. J. 11 (1964), 183–187. MR 166234
Martin Mathieu and Armando R. Villena, The structure of Lie derivations on $C^\ast$-algebras, J. Funct. Anal. 202 (2003), no. 2, 504–525. MR 1990536, DOI 10.1016/S0022-1236(03)00077-6
C. Robert Miers, Lie derivations of von Neumann algebras, Duke Math. J. 40 (1973), 403–409. MR 315466
O. Panaia, Quasi–spatiality of isomorphisms for certain classes of operator algebras, Ph. D. dissertation, University of Western Australia, 1995.
Gordon A. Swain, Lie derivations of the skew elements of prime rings with involution, J. Algebra 184 (1996), no. 2, 679–704. MR 1409235, DOI 10.1006/jabr.1996.0281
Gordon A. Swain and Philip S. Blau, Lie derivations in prime rings with involution, Canad. Math. Bull. 42 (1999), no. 3, 401–411. MR 1703701, DOI 10.4153/CMB-1999-047-6
A. R. Villena, Lie derivations on Banach algebras, J. Algebra 226 (2000), no. 1, 390–409. MR 1749896, DOI 10.1006/jabr.1999.8193