Characterizations of Lie derivations on Kadison-Singer algebras

Kadison-Singer algebra (KS-algebra) is a new class of non-self-adjoint operator alge-bras. In this paper, we mainly study the standardization of Lie derivations on some KS-algebras. In Sect. 2, we prove that if L is a non-trivial KS-lattice in M-3(C) , then every Lie derivation from AlgL into M-3(C) is standard. In Sect. 3, we suppose that H is a separable infinite-dimensional Hilbert space, N is a non-trivial nest on H and ? is a separating vector for N". Let L be a KS-lattice generated by N and the rank-one projection P-? , we prove that every Lie derivation from AlgL into B(H) is standard.