Unbounded operators
Allan M. Sinclair, Roger F. Smith · 2008
Introduction For the most part, these notes are concerned with the bounded operators which constitute the von Neumann algebras under consideration. However, results from the theory of unbounded operators have played a role in Chapter 9, and knowledge of this topic is essential for reading the literature in this area. Since we feel that this theory is less well known than its counterpart for bounded operators, we include here a brief exposition of the main theorems required in these notes. Most of what is needed may be found in [104, Section 5.6], and we follow their development to a considerable extent. However, we have specific goals for the theory and we do not offer a comprehensive treatment. The main objective is to understand the operators that arise as unbounded left multiplication operators on II 1 factors. Section B.2 contains the basic theory of closed and closable operators. In Section B.3, we develop as much of the functional calculus as we will need. We carry this out for positive operators, relating matters to the well known functional calculus for bounded positive operators. Along the way we establish the polar decomposition of a closed operator which appeared in Lemma 9.4.2. The important unbounded operators of II 1 factor theory are those that arise from vectors in L 2 ( N ) and L 1 ( N ), and we lay out their theory in Sections B.4 and B.5 respectively.