Mackey imprimitivity and commuting tuples of homogeneous normal operators

In this semi-expository article, we investigate the relationship between the imprimitivity introduced by Mackey several decades ago and commuting d- tuples of homogeneous normal operators. The Hahn–Hellinger theorem gives a canonical decomposition of a \(*\) $\ast$ - algebra representation \(\rho \) $\rho$ of \(C_0({\mathbb {S}})\) $C_0(\mathbb{S})$ (where \({\mathbb {S}}\) $\mathbb{S}$ is a locally compact Hausdorff space) into a direct sum. If there is a group G acting transitively on \({\mathbb {S}}\) $\mathbb{S}$ and is adapted to the \(*\) $\ast$ - representation \(\rho \) $\rho$ via a unitary representation U of the group G, in other words, if there is an imprimitivity, then the Hahn–Hellinger decomposition reduces to just one component, and the group representation U becomes an induced representation, which is Mackey’s imprimitivity theorem. We consider the case where a compact topological space \(S\subset {\mathbb {C}}^d\) $S\subset {\mathbb{C}}^d$ decomposes into finitely many G- orbits. In such cases, the imprimitivity based on S admits a decomposition as a direct sum of imprimitivities based on these orbits. This decomposition leads to a correspondence with homogeneous normal tuples whose joint spectrum is precisely the closure of G- orbits.