|XXXIV Workshop on Geometric Methods in Physics||28.06-4.07.2015|
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Quantization and $C^*$-algebras of nilpotent Lie groups
We will describe the image of operator valued Fourier transforms on
general nilpotent Lie groups. Our approach relies on suitable stratifications of the duals of nilpotent Lie algebras and a Lie theoretic method of quantization of coadjoint orbits based on global canonical coordinates. We thus show that the $C^*$-algebra of any nilpotent Lie group is a solvable $C^*$-algebra with special spectral properties. A characterization of Heisenberg groups in terms of group
$C^*$-algebras will be provided along these lines.
The talk is based on joint work with Ingrid Beltita and Jean Ludwig.