Physics – Mathematical Physics
Scientific paper
1998-04-02
Int.J.Mod.Phys. A14 (1999) 129-146
Physics
Mathematical Physics
24 pages, Latex
Scientific paper
10.1142/S0217751X99000063
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.
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