The focus of this dissertation is the study of Kadison-Singer algebras. We show that many Kadison-Singer algebras are maximal triangular in all algebras containing them although their definition requires the maximality taken in the class of reflexive algebras. We have classified diagonal-trivial maximal non selfadjoint subalgebras of matrix algebras with lower dimensions. We show that Alg(F3) is maximal in the class of reflexive subalgebras of B( H), where the restriction of the diagonal of each such subalgebra in LG3 is trivial. We show that the reflexive algebras Alg(L) given by a double triangular lattice L in M (a factor of type II1) is maximal non selfadjoint in the class of all weak operator closed subalgebras with respect to its diagonal if L generates M. Our method can be used to prove similar results in finite dimensional matrix algebras. As a consequence, we give a new proof to Hou's result, that is, each reflexive algebra determined by a double triangular lattice of projections in a matrix algebra has the diagonal maximality if the double triangular lattice of projections generated the whole matrix algebra.
|Commitee:||Ge, Liming, Hadwin, Don, Hibschweiler, Rita, Hinson, Edward, Shen, Junhao, Zhang, Yitang|
|School:||University of New Hampshire|
|School Location:||United States -- New Hampshire|
|Source:||DAI-B 76/06(E), Dissertation Abstracts International|
|Keywords:||Double triangle lattice, Factor of type II1, Kadison-singer algebras, Kadison-singer lattice, Reflexive algebra|
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