Doctoral Dissertations

Fall 2009

Dissertation

Mathematics

Degree Name

Doctor of Philosophy

Liming Ge

Abstract

In this dissertation, we defined a new class of non selfadjoint operator algebras---Kadison-Singer algebras or KS-algebras for simplicity. These algebras combine triangularity, reflexivity and von Neumann algebra property into one consideration. Generally speaking, KS-algebras are reflexive, maximal triangular with respect to its "diagonal subalgebra". Many selfadjoint features are preserved in them and concepts can be borrowed directly from the theory of von Neumann algebras. In fact, a more direct connection of KS-algebras and von Neumann algebras is through the lattice of invariant projections of a KS-algebra. The lattice is reflexive and "minimally generating" in the sense that it generates the commutant of the diagonal as a von Neumann algebra.

This dissertation consists of three chapters. In chapter 1, we give some background and the definition of Kadison-Singer algebras (as well as corresponding Kadisalong with some basic properties of KS-algebras). In chapter 2, we construct Kadision-Singer factors with hyperfinite factors as their diagonals, study their commutant and describe the corresponding Kadison-Singer lattices in details. At the end, a lattice invariant is introduced to distinguish these lattices. In chapter 3, we first review the results of reflexive algebras determined by two projections, then describe the reflexive lattice generated by three free projections and show that it is a Kadison-Singer lattice and thus the corresponding algebra is a Kadison-Singer algebra. We also show that this lattice is homeomorphic to two-dimensional sphere S2 (plus two distinct points corresponding to 0 and I). Then we introduce a notation of connectedness of projections in a lattice of projections in a finite von Neumann and show that all connected components form another lattice, called a reduced lattice. Reduced lattices of most of our examples were computed. We end this dissertation by discussing maximal triangularity in different aspects.

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