Conditional Expectations and Projection Maps of von Neumann Algebras

Abstract

Conditional expectations have been around in operator algebras for more than 40 years. They seem to be first studied by Umegaki [26] in 1954 and have been of increasing importance since then, especially after subfactor theory became a major topic in von Neumann algebras. There is no systematic treatment of the theory in the literature; the results are scattered appearing often as lemmas needed for other theorems. The present note will partly compensate for this in that its first part will be a rudimentary survey of the theory with the aim of classifying or rather relating conditional expectations to well-known special cases. In the second part we shall study the more general class of projection maps, i.e. idempotent, unital positive linear maps of von Neumann algebras into themselves. Such maps are conditional expectations if and only if they are completely positive, and their images are Jordan algebras instead of von Neumann algebras. We shall see that their theory is intimately related to conditional expectations onto the von Neumann algebra generated by the image.