Kadison-Singer Algebras And Relative Problems

In this paper,there are four main chapters.This paper introduces the development of operator algebras,operator theories and Kadison-Singer algebras briefly in the first section.We also give some basic concepts.Besides,the critical problems which are discussed in the paper are discribed.Then,this paper constructs a new class of pentagon subspace lattices L={(0),L,M,K,H} whose gap dimension is one,satisfying M= L v {Cx0},where H is a complex Hilbert space and the nonzero vector x0 belongs to K?,but does not belong to K+ L.We prove that AlgL is a semi-simple Kadison-Singer algebra.Besides,this paper proposes another constructional technique of Kadison-Singer algebras in the third section.A maximal nest and a general nest are spanned by a point respectively by Wang Liguang[21]and Hou Chengjun[3],thus,we get a special class of Kadison-Singer algebras by proving that the lattices spanned by nest are generated minimally.And we construct a simple example.Finally,this paper discusses the similarity relation between CSL algebras which are not nest algebras and Kadison-Singer algebras,and give two examples of CSL algebras,not nest algebras,which are subalgebras of M3(C).Let A =(?),and by calculating,this paper explains that A can be similar to a Kadison-Singer algebra,but B never.