On the Diagonals of Projections in Matrix Algebras Over Von Neumann Algebras

The main focus of this dissertation is on exploring methods to characterize the diagonals of projections in matrix algebras over von Neumann algebras. This may be viewed as a non-commutative version of the generalized Pythagorean theorem and its converse (Carpenter's Theorem) studied by R. Kadison. A combinatorial lemma, which characterizes the permutation polytope of a vector in $\mathbb{R}^n$ in terms of majorization, plays an important role in a proof of the Schur-Horn theorem. The Pythagorean theorem and its converse follow from this as a special case. In the quest for finding a non-commutative version of the lemma alluded to above, the notion of C*-convexity looks promising as the correct generalization for convexity. We make generalizations and improvements of some results known about C*-convex sets. We prove the Douglas lemma for von Neumann algebras and use it to prove some new results on the one-sided ideals of von Neumann algebras. As a useful technical tool, a non-commutative version of the Gram-Schmidt process is proved for finite von Neumann algebras. A complete characterization of the diagonals of projections in full matrix algebras over an abelian C*-algebra is provided in chapter 5. In chapter 6, we study the problem in the case of M_2(M_n(C)), the full algebra of 2 x 2 matrices over M_n(C). The example gives us hints regarding the possibility of extracting an underlying notion of convexity for C*-polytopes, which are not necessarily convex.