Banach空间上一类套代数的李环同构

Lie Ring Isomorphisms between Certain Nest Algebras on Banach Spaces

令N,M分别是(实或复)数域F上的Banach空间X和Y上的套,具有性质:(0)和X都是N的极限点,即(0)+=(0),X-=X.令AlgN和AlgM分别为相应的套代数。证明了映射Φ:AlgN→AlgM是李环同构(即Φ是可加、李可乘的双射)当且仅当Φ(A)=TAT-1+h(A)I对任意的A∈AlgN都成立,或Φ(A)=-TA*T-1+h(A)I对任意的A∈AlgN都成立,其中h是在所有交换子上为零的可加泛函,T是可逆的有界线性或共轭线性算子。

Let N and M be nests on Banach spaces X and Y over the （real or complex） field F, with the property that both （0） and X are limit points of N, i. e. , （0）＋ = （0） and X- =X. Let Alg N and AlgM be the associated nest algebras, respectively. It is shown that a map Ф：AlgN→AlgM is a Lie ring isomorphism （i. e. , 69 is additive, Lie multiplicative and bijective） if and only if Ф（A）=TAT^-1＋h（A）I for all A∈ AlgN or Ф（A） = -TA^＊ T^-1 ＋h（A）I for all A∈ AlgN, where h is an additive functional vanishing on all commutators and T is an invertible bounded linear or conjugate linear operator.