These notes are an introduction to some of the theory of finite von Neumann algebras and their von Neumann subalgebras, with the emphasis on maximal abelian self-adjoint subalgebras (usually abbreviated masas). Assuming basic von Neumann algebra theory, the notes are fairly detailed in covering the basic construction, perturbations of von Neumann subalgebras, general results on masas and detailed ones on singular masas in II1 factors. Due to the large volume of research on finite von Neumann algebras and their masas the authors have been forced to be selective of the topics included. Nevertheless, a substantial body of recent research has been covered.

Each chapter of the book has its own introduction, so the overview of the contents below will be quite brief. We have also included a discussion of a few important results which have been omitted from the body of the text. In each case, we felt that the amount of background required for a reasonably self-contained account was simply too much for a book of this kind.

We have tried to make the material accessible to graduate students who have some familiarity with von Neumann algebras at the level of a first course in the subject. The early chapters review some of this, but are best read by the beginner with one of the standard texts, [104, 105, 187], to hand to fill in any gaps.