Cohomology Of A Class Of Kadison-Singer Algebras

The paper discuss the problem that the properties and the cohomology groups of the Kadison-Singer algebras. Specifically, Suppose L is a subspace lattice. In with A is a n-order triangular matrix and the row sums of A are all equal, B is a n-order matrix}.1-th bounded coho-mology group H1(Alg(L), Alg(L)) of the algebras Alg(L) with coefficients in Alg(L) is ordinary.that is, its every bounded linear derivation from Alg(L) into the alge-bras itself is inner. And all n-th bounded cohomology groups Hn(Alg(L),M2n(C)) of Alg(L) with coefficients in M2n(C) are ordinary for all n≥1. In addition, we have the conclusion of the basic features properties of the KS-algebras. such as the dimension, the base and the Center. This paper is divided into three chapters.In section 1, we give an introduction, firstly we describe the background and the preliminary knowledge of Kadison-Singer algebras and cohomology groups on operator algebras detailedly. In addition, we present the KS-algebras of this article simply. Then we introduce the problem that whether every bounded linear derivation from Alg(L) into itself is inner derivation and propose whether all n-th bounded cohomology groups of Alg(L) with coefficients in M2n(C) are ordinary for all n≥1. This is the Central Question of the study in this paper.In section 2, we study some basic properties of the Alg(L), it depicts some basic features of the algebras, such as the dimension, the base and the centerIn section 3, it is the major part of this dissertation.we mainly investigate the question whether the 1-th bounded cohomology group is ordinary, that is, we prove that every bounded linear derivation from Alg(L) into itself is inner derivation, and show that all n-th bounded cohomology group of Alg(L) with coefficients in M2n(C) are ordinary for all n≥1.