PROJECTIVE GROUP ALGEBRAS

Abstract

In this paper we apply a recently proposed algebraic theory of integration to projective group algebras. These structures have received some attention in connection with the compactification of the M theory on noncommutative tori. This turns out to be an interesting field of applications, since the space [Formula: see text] of the equivalence classes of the vector unitary irreducible representations of the group under examination becomes, in the projective case, a prototype of noncommuting spaces. For vector representations the algebraic integration is equivalent to integrate over [Formula: see text]. However, its very definition is related only at the structural properties of the group algebra, therefore it is well defined also in the projective case, where the space [Formula: see text] has no classical meaning. This allows a generalization of the usual group harmonic analysis. Particular attention is given to Abelian groups, which are the relevant ones in the compactification problem, since it is possible, from the previous results, to establish a simple generalization of the ordinary calculus to the associated noncommutative spaces.