Convergence properties of bimodules over maximal abelian self-adjoint algebras

• Main Author: Habgood, Joseph Letuya
• Published: Queen's University Belfast 2007
• Subjects: 510
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.486216

The study of structure of a topological natur~ on the collection of von Neumann algebras was initiated in the 1960s. Since then, the topic has attracted the attention of a number of authors and several deep results have been established. A key result in this area is that the map sending a von Neumann algebra to its commutant is continuous. In 1990 Shulman used a notion of convergence on certain classes of non-self-adjoint operator algebras to show that certain subspace lattices are reflexive. These results were extended by Shulman and Todorov in 2004. The main theorems in these works are continuity results for the map sending a unital operator algebra A to the lattice of invariant subspaces latA. In this thesis continuity results are established in the context of ~asa-bimodules. It is shown that the map sending a masa-bimodule to its support is continuol,ls. This uses a generalisation of a continuity of lat result together with some new techniques that are developed in this thesis, which are based on a generalisation of the capacity introduced by Haydon and Shulman in 1996. A novel aspect of the work in this thesis is that the continuity of the maps which send an w-closed set to the maximal (resp. minimal) weak* closed masa-bimodule supported on that set are also considered. The existence of non-synthetic w-closed sets is used to show that these maps are not continuous, and semi-continuity results are given for both maps. In some special cases much more complete results are obtained. The work in early chapters of this thesis is largely to develop new tools that facilitate the proofs of the continuity results.