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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 $A, B\in\mathrm{Alg}\mathcal {N}$.

47L35Nest algebras, CSL algebras
16W10Associative rings with involution, etc.
Full Text: DOI
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