Fredholm theory in Von Neumann algebras

Abstract

The main goal of this study is to generalize the theory of compact and of Fredholm operators defined on a complex Hilbert space H to von Neumann algebras. Since this generalization depend heavily on the study of the project ion lattice existing on a von Neumann algebra, the first chapter contains a comprehensive amount of standard material concerning the geometry of projections in a von Neumann algebra A. If we consider the commutant .A.' of a von Neumann algebra and a projection E in .A. then the restriction of each element of .A.' to E(H) defines a representation HE of .A.' into the C* - algebra of all bounded linear operators on E(H) (E(H) is the range space of the projection E). In Chapter 2 we consider all these representations of .A. ' into E ( H) ( where E is assumed to be finite relative to .A.), to construct a commutative monoid M. The Grothendieck group r of M can canonically be equipped with an order relation. This group is important in the Chapters that follow, since it contains the so called indices of the Fredholm elements defined on a von Neumann algebra .A. In Chapter 3 the concept of finite, compact and Fredholm elements are introduced. On the set of all Fredholm elements relative to .A. an index mapping is defined with values in the Grothendieck group r. These values are called the indices of the Fredholm elements relative to .A. The main theorems of this study are obtained in Chapter 4. These results generalize theorems, obtained by F. Riesz and Atkinson.