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Characterization of Lie multiplicative isomorphisms between nest algebras. (English) Zbl 1273.47123
Let $\mathcal {A}$ and $\mathcal {A}^{\prime }$ be two algebras. A map $\Phi :\mathcal {A}\to \mathcal {A}^{\prime }$ is called a Lie multiplicative map if $\Phi ([A, B])=[\Phi (A), B]+[A, \Phi (B)]$ for any $A, B\in \mathcal {A}$, where $[A, B]=AB-BA$ is the usual Lie product. Moreover, if $\Phi$ is both bijective and linear additionally, then $\Phi$ is called a Lie isomorphism. This paper is devoted to the characterization of Lie multiplicative maps on nest algebras. Let ${\text{Alg}}\mathcal{N}$ and $\text{Alg}\mathcal{M}$ be two nest algebras, where $\mathcal{N}$ and $\mathcal{M}$ are nests in Banach spaces $X$ and $Y$ over the real or complex field $\mathbb{F}$, respectively, with the property that, if $M \in \mathcal{M}$ such that $M _{ - } = M$, then $M$ is complemented in $Y$. Assume that there is a nontrivial element in $\mathcal{N}$ which is complemented in $X$ and $\Phi :\text{Alg}\mathcal{N} \to \text{Alg}\mathcal{M}$ is a bijective map. Then $\Phi$ is Lie multiplicative if and only if $\Phi$ is of the form $\Psi +\tau$, where $\Psi :\text{Alg}\mathcal{N} \to \text{Alg}\mathcal{M}$ is a ring isomorphism or the negative of a ring anti-isomorphism and $\tau:\mathrm{Alg}\mathcal {N}\to \mathbb {F}I$ is a map satisfying $\tau ([A, B])=0$ for all $A, B\in\mathrm{Alg}\mathcal {N}$.

##### MSC:
 47L35 Nest algebras, CSL algebras 16W10 Associative rings with involution, etc.
Full Text:
##### References:
 [1] Bai Z F, Du S P, Hou J C. Multiplicative Lie isomorphisms between prime rings. Comm Algebra, 2008, 36: 1626--1633 · Zbl 1145.16013 · doi:10.1080/00927870701870475 [2] Beidar K I, Bresar M, Chebotar M A, et al. On Herstein’s Lie map conjectures (I). Trans Amer Math Soc, 2001, 353: 4235--4260 · Zbl 1019.16019 · doi:10.1090/S0002-9947-01-02731-3 [3] Beidar K I, Bresar M, Chebotar M A, et al. On Herstein’s Lie map conjectures (III). J Algebra, 2002, 249: 59--94 · Zbl 1019.16021 · doi:10.1006/jabr.2001.9076 [4] Beidar K I, Martindale III W S, Mikhalev A V. Lie isomorphisms in prime rings with involution. J Algebra, 1994, 169: 304--327 · Zbl 0813.16020 · doi:10.1006/jabr.1994.1286 [5] Berenguer M I, Villena A R. Continuity of Lie isomorphisms of Banach algebras. Bull London Math Soc, 1999, 31: 6--10 · Zbl 1068.46500 · doi:10.1112/S0024609398005128 [6] Bresar M. Commuting traces of biadditive mappings, commutativity preserving mappings, and Lie mappings. Trans Amer Math Soc, 1993, 335: 525--546 · Zbl 0791.16028 [7] Davidson K R. Nest Algebras. Pitman Research Notes in Mathematics, vol. 191. London-New York: Longman, 1988 [8] Hou J C, Zhang X L. Ring isomorphisms and linear or additive maps preserving zero products on nest algebras. Linear Algebra Appl, 2004, 387: 343--360 · Zbl 1061.47035 · doi:10.1016/j.laa.2004.02.032 [9] Marcoux L W, Sourour A R. Lie isomorphisms of nest algebras. J Funct Anal, 1999, 164: 163--180 · Zbl 0940.47061 · doi:10.1006/jfan.1999.3388 [10] Martindale III W S. Lie isomorphisms of prime rings. Trans Amer Math Soc, 1969, 142: 437--455 · Zbl 0192.37802 · doi:10.1090/S0002-9947-1969-0251077-5 [11] Martindale III W S. Lie isomorphisms of simple rings. J London Math Soc, 1969, 44: 213--221 · Zbl 0164.03901 [12] Qi X F, Hou J C. Additivity of Lie multiplicative maps on triangular algebras. Linear Multilinear Algebra, in press · Zbl 1216.47112 [13] Radjavi H, Rosenthal P. Invariant Subspaces. Berline-Heidelberg-New York: Springer-Verlag, 1973 · Zbl 0269.47003