The covariant Stone–von Neumann theorem for locally compact quantum groups

Abstract

The Stone–von Neumann theorem is a fundamental result which unified the competing quantum-mechanical models of matrix mechanics and wave mechanics. In this article, we continue the broad generalization set out by Huang and Ismert and by Hall, Huang, and Quigg, analyzing representations of locally compact quantum-dynamical systems defined on Hilbert modules, of which the classical result is a special case. We introduce a pair of modular representations which subsume numerous models available in the literature and, using the classical strategy of Rieffel, prove a Stone–von Neumann-type theorem for maximal actions of regular locally compact quantum groups on elementary C*-algebras. In particular, we generalize the Mackey–Stone–von Neumann theorem to regular locally compact quantum groups whose trivial actions on \(\mathbb {C}\) are maximal and recover the multiplicity results of Hall, Huang, and Quigg. With this characterization in hand, we prove our main result showing that if a dynamical system \((\mathbb {G},A,\alpha )\) satisfies the multiplicity assumption of the generalized Stone–von Neumann theorem, and if the coefficient algebra A admits a faithful state, then the spectrum of the iterated crossed product \(\widehat{\mathbb {G}}^\textrm{op}\ltimes (\mathbb {G}\ltimes A)\) consists of a single point. In the case of a separable coefficient algebra or a regular acting quantum group, we further characterize features of this system, and thus obtain a partial converse to the Stone–von Neumann theorem in the quantum group setting. As a corollary, we show that a regular locally compact quantum group satisfies the generalized Stone–von Neumann theorem if and only if it is strongly regular.