矩阵代数的Kadison-Singer格的分类

Classification of Kadison-Singer Lattices in Matrix Algebras

研究了矩阵代数M_n(C)的KS格,证明了每个生成M_3(C)的KS格都相似于(?)_0或I-(?)_0,其中(?)_0为M_3(C)的一个极大对角投影套和一个赋值全非零的秩1投影所生成的KS格,从而M_3(C)的对角平凡的KS代数都是4维的.同时,还给出了几个生成M_4(C)但非同构的KS格的例子.

We study Kadison-Singer lattices in the matrix algebra M_n（C）,and prove that each Kadison-Singer lattice generating M_3（C） as an algebra is similar to L_0 or I-L_0,where L_0 is the KS lattice generated by a maximal nest of diagonal projections and a rank one projection matrix with nonzero entries in M_3（C）,hence each Kadison-Singer algebra with trivial diagonal in M_3（C） has dimension 4.In addition,we give some examples of nonisomorphic Kadison-Singer lattices which generate M_4（C）.