Structure of the $C^*$-Algebras of Nilpotent Lie Groups

Abstract

We show that the algebraic structure of the group $C^*$-algebra $C^*(G)$ of a simply connected, connected nilpotent Lie group $G$ is described as repeating finitely the extension of $C^*$-algebras with $T_{2^-}$ spectrums by themselves and one more extension by a commutative $C^*$-algebra on the fixed point space $(\mathfrak{G}^*)^G$ of $\mathfrak{G}^*$ under the coadjoint action of $G$. Using this result, we show that $C^*(G)$ has no non-trivial projections.