Topological Groups, Invariant Measures, Convolutions and Representations

Abstract

This chapter is almost independent of the previous ones and it can be read accordingly. For the first four sections only elementary point set theoretical and measure theoretical background is presupposed. Thus Section 1 can be read immediately and before starting Sections 2, 3 or 4 it is sufficient to be familiar with the contents of Section III.7. The remaining two sections require a casual acquaintence with the fundamental concepts of representations in Hilbert spaces. A partial reading of Sections IV.1, 2 and 3 is satisfactory for this purpose. In Section 5 we shall use also some basic facts about v. Neumann algebras A. In particular we need that A is its second commutant and that A is weakly closed. All of this is discussed and proved in Section III.6. The text contains detailed references to this background material at the places where it is used. See also the preliminary remarks on the last page.