∗ A Generalization of the Stone-von Neumann Theorem

Abstract

The relevance of non-regular representations of the Heisenberg group (or of the Weyl C∗-algebra \(\mathcal{A}_{W}\)) raises the question of a possible classification of them, which generalizes Stone-von Neumann (SvN) theorem. For this purpose, a possible strategy is to consider a maximal abelian \(\mathcal{A}\) subalgebra of \(\mathcal{A}_{W}\), identify its Gelfand spectrum \(\Sigma (\mathcal{A})\) and classify the realizations of such an abelian algebra in terms of multiplication operators on \(L^{2}(\Sigma (\mathcal{A}),d\mu )\), with d μ a (Borel) measure on \(\Sigma (\mathcal{A})\).