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Actions exponentielles et idéaux premiers

Abstract : Let G be a nilpotent, connected, simply connected Lie group, whose Lie algebra g is an exponential d-module, if d denotes an exponential algebra of derivations of the Lie algebra g. Let D be the connected, simply connected Lie group with Lie algebra d. One has the following results : the maximal d-invariant ideals in the convolution alegbra of integrable functions on g (resp. in the Schwartz algebra of g) coincide with the kernels of the closed d-orbits of the dual of g (resp. with the restrictions of these kernels to the Schwartz algebra). The proper closed d-prime ideals in the convolution algebra of integrable functions (resp. the proper d-prime ideals in the Schwartz algebra which are closed in the topology induced by an arbitrary Schwartz norm) coincide with the kernels of not necessarily closed d-orbits (resp. with the restrictions of these kernels to the Schwartz algebra). One has the equivalent of the Wiener property for D-invariant ideals. For a closed d-orbit, the kernel of the orbit modulo the closed minimal ideal associated to it (in the convolution algebra of integrable functions) is a nilpotent algebra. For a closed d-orbit the restriction of the kernel to the Schwartz algebra is dense in the kernal itself. Moreover, the closure of every d-orbit contains a closed d-orbit. These results generalize well known properties of nilpotent Lie groups
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Catherine Molitor. Actions exponentielles et idéaux premiers. Mathématiques générales [math.GM]. Université Paul Verlaine - Metz, 1996. Français. ⟨NNT : 1996METZ020S⟩. ⟨tel-01777131⟩



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