Translation group and spectrum condition (Q1066432)

Let A be a \(C^*\)-algebra on which \({\mathbb{R}}^ d\) acts as a symmetry group. Give a closed cone \(\hat V\) in the dual of \({\mathbb{R}}^ d\). The representations of A are characterized having the properties: (i) The authomorphisms \(\alpha_ a\), \(a\in {\mathbb{R}}^ d\) are weakly inner, (ii) the corresponding unitary group representation U(a) is continuous and (iii) the spectrum of \(U(a)\) is contained in \(\hat V\).