Abelian twisted groupoid extensions

This article gives a decomposition of the C*-algebra of a twisted locally compact groupoid with Haar system in presence of a normal sub-groupoid over which the twist is abelian. One obtains then an equivariant Gelfand isomorphism onto another twisted groupoid C*-algebra. This is ap-plied to C*-algebraic deformation quantization and to multiplier representa-tions of locally compact abelian groups.