Kadison-Singer Algebras And Its Characterization Of Linear Mappings

In this paper,we mainly study the structure and properties of some linear mappings on Kadison-Singer algebras.The algebras as that we study include von Neumann algebras,nest algebras and Kadison-Singer algebras.The mappings that we study include Hochschild cohomology group,derivations,inner derivations and Lie derivations.Firstly,we study the structure of Kadison-Singer algebras on finitedimensional Hilbert space and we prove that if L is a non-trivial KadisonSinger lattice of Mn(C)(n≥2),then the corresponding Kadison-Singer algebra AlgL is a non-selfadjoint operator algebra.Secondly,we study the cohomology theory on Kadison-Singer algebras,we prove that if A?M3(C)is a Kadison-Singer algebra,then every n-th Hochschild cohomology group is trivial,that is Hn(AlgL,M3(C))={0}.In particular,when n=1,then every derivation is an inner derivation.Finally,we mainly study the standardization of Lie derivations on KadisonSinger algebras,we prove that if L is a non-trivial Kadison-Singer lattice in M3(C),then every Lie derivation from AlgL into M3(C)is standard.In addition,we prove that if is an infinite-dimensional separable Hilbert space,N is a non-trivial nest on H and ξ is a separating vector for N".Let L be a Kadison-Singer lattice generated by N and the rank-one projection Pξ,then every Lie derivation from AlgL into B(H)is standard.