The $C^*$-Algebra of the Heisenberg Motion Groups ${{\mathbb {T}}}^n\ltimes {\mathbb {H}}_n$

Abstract

We show that the tensor product $A\otimes B$ of two $C^*$-algebras satisfying the norm controlled dual limits (NCDL) conditions has again the same property. We use this result to describe the $C^*$-algebra of the Heisenberg motion groups $G_n={{\mathbb {T}}}^n\ltimes {\mathbb {H}}_n$ as algebra of operator fields defined over the spectrum of $G_n$. The main problem for this description is to find the explicit NCDL-conditions for the group $G_1$: Let ${{\mathcal {F}}}:C^*(G_1)\rightarrow l^{\infty }(\widehat{G_1})$ be the operator valued Fourier transform of $G_1$. If $(\pi _k)_{k\in {\mathbb {N}}}$ is a properly converging sequence in the spectrum of $G_1$ which converges in the Fell topology to a limit set L with more than one element, we must understand the behaviour at infinity of the sequence of bounded linear operators $(\pi _k(a))_{k\in {\mathbb {N}}}$ for every $a\in C^*(G_1)$.