Abelian Automorphism Group

Abstract

This chapter is devoted to the spectral analysis of actions of abelian groups on a von Neumann algebra which will be used in the subsequent chapters. As a Banach space, an operator algebra presents a great challenge to functional analysis. Although an operator algebra lives on a Hilbert space, it behaves very pathologically from the point of view of Banach spaces. For example, the spectral decomposition of a one parameter group of automorphisms on a C*-algebra is out of question. Unless the group is periodic, any attempt of decomposition of the algebra relative to spectrum has been defeated. Thus, we take a moderate approach by examining the concept of spectrum of an element of the algebra relative to a given action of a locally compact abelian group. To this end, first we will look at an abelian group action on a Banach space and then move to analysis of automorphism actions of such a group on a von Neumann algebra M. The formal definition of the spectrum Sp _α (x) of an element of x ∈ M is defined in somewhat convoluted way. But it is, roughly speaking, nothing but the simple oscillation component of the function: s ∈ G ↦ α _s (x) ∈ M. The reader should make the techniques in this chapter as reliable tools. For those readers who are not comfortable with general locally compact groups, it is advised to assume that all the groups involved are the additive group R of real numbers.