Abstract
In this paper we prove that every bijection preserving Lie products from a triangular algebra onto a normal triangular algebra is additive modulo centre. As an application, we described the form of bijections preserving Lie products on nest algebras and block upper triangular matrix algebras.
References
[1] M. Brešar, Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. Amer. Math. Soc. 335 (1993) 525-546.10.1090/S0002-9947-1993-1069746-XSearch in Google Scholar
[2] D. Benkovič, D. Eremita, Commuting traces and commutativity preserving maps on triangular algebras, J. Algebra, 280 (2004) 797-824.Search in Google Scholar
[3] D. Benkovič, Biderivations triangular algebras, Linear Algebra Appl. 431 (2009) 1587-1602.10.1016/j.laa.2009.05.029Search in Google Scholar
[4] M. Brešar, P. Šemrl, Commutativity preserving linear maps on central simple algebras, J. Algebras, 284 (2005) 102-110.Search in Google Scholar
[5] M. Choi, A. Jafarian, H. Radjavi, Linear maps preserving commutativity, Linear Algebra Appl. 87 (1987) 227-241.10.1016/0024-3795(87)90169-8Search in Google Scholar
[6] W.S. Cheung, Commuting maps of triangular algebras, J. London math. Soc. 63 (2001) 117-127.10.1112/S0024610700001642Search in Google Scholar
[7] W.S. Cheung, Lie derivation of triangular algebras, Linear Multilinear Algebra, 51 (2003) 299-310.Search in Google Scholar
[8] K.R. Davidson, Nest algebras, Pitman Research Notes in Mathematics Series, Longman Scientific and Technical, 1988.Search in Google Scholar
[9] J.C. Hou, M.Y. Jiao, Additive maps preserving Jordan zero-products on nest algebras, Linear Algebra Appl. 429 (2008) 190-208.10.1016/j.laa.2008.02.021Search in Google Scholar
[10] F.Y. Lu, Additive Jordan isomorphisms of nest algebras on normed spaces, J. Math. Anal. Appl. 284 (2003) 127-143.10.1016/S0022-247X(03)00287-7Search in Google Scholar
[11] L.W. Marcoux, Lie isomorphisms of nest algebras, J. Funct. Anal. Appl. 164 (1999) 163-180.10.1006/jfan.1999.3388Search in Google Scholar
[12] W.S. Martindale, Lie isomorphisms of simple rings, J. London Math. Soc. 44 (1969) 213-221.10.1112/jlms/s1-44.1.213Search in Google Scholar
[13] C.R. Miers, Commutativity preserving maps of factors, Canad. J. Math. 40 (1988) 248-256.10.4153/CJM-1988-011-1Search in Google Scholar
[14] C.R. Miers, Lie isomorphisms of operator algebras, Pacific J. Math. 38 (1971) 717-735.10.2140/pjm.1971.38.717Search in Google Scholar
[15] C.R. Miers, Lie isomorphisms of factors, Trans. Amer. Math. Soc. 147 (1970) 55-63.10.1090/S0002-9947-1970-0273423-7Search in Google Scholar
[16] L. Molnár, P. Šemrl, Nonlinear commutativity preserving maps on self-adjoint operators, Q. J. Math. 56 (2005) 589-595.10.1093/qmath/hah058Search in Google Scholar
[17] M. Omladič, H. Radjavi, P. Šemrl, Preserving commutativity, J. Pure Appl. Algebra 156 (2001) 309-328.10.1016/S0022-4049(99)00154-1Search in Google Scholar
[18] P. Šemrl, Non-linear commutativity preserving maps, Acta Sci. Math. (Szeged) 71 (2005) 781-819.Search in Google Scholar
[19] T.L. Wong, Jordan isomorphisms of triangular algebras, Linear Algebra Appl. 418 (2006) 225-233.10.1016/j.laa.2006.02.001Search in Google Scholar
[20] J.H. Zhang, W.Y. Yu, Jordan derivations of triangular algebras, Linear Algebra Appl. 419 (2006) 251-255.10.1016/j.laa.2006.04.015Search in Google Scholar
[21] W.Y. YU, J.H. Zhang, Nonlinear Lie derivations of triangular algebras, Linear Algebra Appl. 432 (2010) 2953-2960.10.1016/j.laa.2009.12.042Search in Google Scholar
[22] W.Y. YU, J.H. Zhang, Lie triple derivations of CSL algebras, Int Theor Phys. 52 (2013) 2118-2127.10.1007/s10773-013-1507-5Search in Google Scholar
[23] J.H. Zhang, F.J. zhang, Nonlinear maps preserving lie products on factor von Neumann algebras, Linear Algebra Appl. 429 (2008) 18-30.10.1016/j.laa.2008.01.031Search in Google Scholar
[24] W.Y. YU, J.H. Zhang, Nonlinear *-Lie derivations on factor von Neumann algebras, Linear Algebra Appl. 437 (2012) 1979-1991.10.1016/j.laa.2012.05.032Search in Google Scholar
© 2016 Weiyan Yu, published by De Gruyter Open
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