Classification Of Kadison-Singer Lattices In4×4Matrix Algebras

In this paper, the author has discussed some Kadison-Singer lattices in4x4ma-trix algebra M4(C). According to the von Neumann algebras generated by these lat-tices in M4(C), we completely classified all Kadison-Singer lattices under the similar-ity which generate abelian von Neumann algebras,M2(C)⊕CI2,M2(C)⊕C⊕Cand M2(C)⊕M2(C) respectively.In the first chapter, we introduced some fundamentals theory of operator algebras and the definition of Kadison-Singer lattice, ect.In the following chapter, according to the von Neumann algebras generated by the Kadison-Singer lattices in M4(C), we classified the kadison-singer lattices which generate abelian von Neumann algebras,M2(C)⊕CI2and M2(C)⊕C⊕Crespectively under the similarity.The third chapter is the main section of this article. We firstly gave some exam-ples of Kadison-Singer lattices in M4(C) which generate the von Neumann algebra M2(C). Following it, we have shown that any Kadion-Singer lattice in which generate M2(C)⊕M2(C) must similar to one of the above-mentioned examples.In the last chapter,we listed some examples on kadison-singer lattices which generated the von Neumann algebra M3(C)⊕C and gave some unsolved questions.