Projective group algebras

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 tor i. This turns out to be an interesting field of applications, since the spa ce (G) over cap of the equivalence classes of the vector unitary irreducibl e representations of the group under examination becomes, in the projective case, a prototype of noncommuting spaces. For vector representations the a lgebraic integration is equivalent to integrate over (G) over cap. However, its very definition is related only at the structural properties of the gr oup algebra, therefore it is well defined also in the projective case, wher e the space (G) over cap has no classical meaning. This allows a generaliza tion of the usual group harmonic analysis. Particular attention is given to Abelian groups, which are the relevant ones in the compactification proble m, since it is possible, from the previous results, to establish a simple g eneralization of the ordinary calculus to the associated noncommutative spa ces.