The \(C^*\)-algebra of the Heisenberg motion groups \(U(d) < imes \mathbb {H}_d.\)

Abstract

Let \(\mathbb {H}_d:=\mathbb {C}^d\times \mathbb {R},\) \((d\in \mathbb {N}^*)\) be the \(2d+1\)-dimensional Heisenberg group and we denote by U(d) (the unitary group) the maximal compact connected subgroup of \(Aut(\mathbb {H}_d),\) the group of automorphisms of \(\mathbb {H}_d.\) Let \(G_d:=U(d) < imes \mathbb {H}_d\) be the Heisenberg motion group. In this work, we describe the \(C^*\)-algebra \(C^*(G_d),\) of \(G_d\) in terms of an algebra of operator fields defined over its dual space \(\widehat{G_d}.\) This result generalizes a previous result in Ludwig and Regeiba (Complex Anal Oper Theory 13(8):3943–3978, 2019).