Satish Pandey, Department of Pure Mathematics, University of Waterloo
"Projections in von Neumann Algebras - Part II”
Continuing with the discussion from last week, we prove as an application of the double commutant theorem that every von Neumann algebra admits a polar decomposition. The concept of Murray-von Neumann equivalence will subsequently be defined for projections in a von Neumann algebra and we will show that the equivalence classes under this relation form a partially ordered set. In the case that the algebra in question is a factor, the set of equivalence classes is in fact totally ordered.