∗ 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 $\Sigma (\mathcal{A})$.