  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
$\hat G$ 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 $\hat G$. 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 $\hat
G$ has no classical meaning. This allows a generalization of the usual group
harmonic analysis. A 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.
