Integration theorems for gages and duality for unimodular groups

Abstract

affiliated with a. with respect to a on (t. In the present work, we investigate analogues of certain portions of ordinary integration theory which Segal did not have occasion to develop in [10], and then we apply the theory of gage spaces to harmonic analysis and duality of unimodular groups. One of the most important ways of constructing an integral in ordinary measure theory is by starting from a positive linear functional on some algebra of bounded functions. In ?1, the corresponding construction of gages is considered. Theorem 1 gives conditions under which a positive central linear functional on a self-adjoint algebra of bounded operators yields a gage on the W*-algebra it generates. This theorem is applied in ?7 to the construction of product gages and in ?9 to the construction of the dual gage for a unimodular group. The question of what can be said about the extension of noncentral positive linear functionals naturally arises. A special case is dealt with in ?2. In [10], convergence nearly everywhere (n.e.) is defined for gage spaces. In ?3, a notion of convergence in measure is introduced. It is found to have the usual relations to convergence n.e. and mean convergence. In ?4, several dominated convergence theorems are proved together with a version of Fatou's lemma. Convergence in measure is most often used in probability spaces. In later work than [10], Segal has used a notion of convergence in probability for self-adjoint operators on a probability gage space, which is defined differently from our convergence in measure. However, the two are