Abstract
LetG be a connected and simply connected solvable Lie group. In a previous paper (cf.[22]) we associated withG a familyM of geometrical objects (“generalized orbits”), and with each elementO ofM a unitary equivalence classF(O) of factor representations. IfG is nilpotent,M coincides with the orbit space of the coadjoint representation, and the mapO→F(O) reproduces essentially the Kirillov isomorphism betweenM and the dual ofG. As a fargoing extension of this, the principal result of the present paper asserts, that upon assigning to 0∈M the kernel of the representation, associated with some element ofF(O), of the groupC * algebraC *(G), we obtain a bijection betweenM and the primitive ideal space ofC *(G).
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Pukanszky, L. The primitive ideal space of solvable Lie groups. Invent Math 22, 75–118 (1973). https://doi.org/10.1007/BF01392298
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DOI: https://doi.org/10.1007/BF01392298