Produits étoile et déformations quantifications strictes

Abstract : We study the structure of the tori equipped with the Rieffel's star product. In particular we determine their spectrum, their center and the type of their factorial representations. This product is like the Moyal product, but the Neumann unicity of representation of commutation relations which says that the regular representation generates a type one factor, is no more true in the case of the tori. Next, we apply the same construction to space of functions on abelian connected groups. We obtain type one-infinity, two-one and two-infinty factors. Then, we try to construct a convergent and tangential deformation quantization on the dual of a nilpotent Lie algebra. Such a construction is realized here in the case of a nilpotent special Lie alegbra, using a deformed version of the formulae of Rieffel. More precisely, we consider a product of exponential, instead of the usual exponential mapping to identify the Lie group with the Lie algebra. Finally we define the adapted Fourier transform and we related it to unitary representations of the corresponding Lie group
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Nabiha Ben Amar. Produits étoile et déformations quantifications strictes. Mathématiques générales [math.GM]. Université Paul Verlaine - Metz, 1992. Français. ⟨NNT : 1992METZ005S⟩. ⟨tel-01775963⟩

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