Characterization of Lie multiplicative isomorphisms between nest algebras

Let N and M be nests on Banach spaces X and Y over the real or complex field F, respectively, with the property that if M ∈ M such that M_ =M, then M is complemented in Y. Let AlgN and AlgM be the associated nest algebras. Assume that Ф ： AlgN → AlgM is a bijective map. It is proved that, if dim X = ∞ and if there is a nontrivial element in N which is complemented in X, then Ф is Lie multiplicative （i.e. Ф（[A, B]） = [Ф（A）, Ф（B）] for all A, B ∈ AlgN） if and only if Ф has the form Ф（A） = TAT^-1 ＋ τ（A） for all A ∈ AlgAN or Ф（A） = -TA^＊T^-1 ＋ τ（A） for all A ∈ AlgN, where T is an invertible linear or conjugate linear operator and τ ： AlgN →FI is a map with τ（[A, B]） = 0 for all A, B ∈ AlgN. The Lie multiplicative maps are also characterized for the case dim X 〈 ∞.

Let N and M be nests on Banach spaces X and Y over the real or complex field F,respectively,with the property that if M∈M such that M-=M,then M is complemented in Y.Let AlgN and AlgM be the associated nest algebras.Assume that Φ:AlgN→AlgM is a bijective map.It is proved that,if dim X=∞ and if there is a nontrivial element in N which is complemented in X,then Φ is Lie multiplicative (i.e.Φ([A,B])=[Φ(A),Φ(B)] for all A,B∈AlgN) if and only if Φ has the form Φ(A)=-TA*T-1+τ(A) for all A∈AlgN or Φ(A)=TAT-1+τ(A) for all A∈AlgN,where T is an invertible linear or conjugate linear operator and τ:AlgN→FI is a map with τ([A,B])=0 for all A,B∈AlgN.The Lie multiplicative maps are also characterized for the case dim X<∞.