Applications of the work of Stone and von Neumann to wavelets

An Infinitesimal Version of the Stone-von Neumann Theorem

In this paper, we present an infinitesimal version of the Stone-von Neumann Theorem. This work was motivated by the need to formulate the uniqueness property of the Heisenberg Commutation Relation purely in terms of unbounded operators.

The Modular Stone-von Neumann Theorem

In this paper, we use the tools of nonabelian duality to formulate and prove a far-reaching generalization of the Stone-von Neumann Theorem to modular representations of actions and coactions of locally compact groups on elementary $ C^{\ast} $-algebras. This greatly extends the Covariant Stone-von Neumann Theorem for Actions of Abelian Groups recently proven by L. Ismert and the second author. Our approach is based on a new result about Hilbert $ C^{\ast} $-modules that is simple to state yet is widely applicable and can be used to streamline many previous arguments, so it represents an impro…

The Covariant Stone-von Neumann Theorem for Actions of Abelian Groups on $ C^{\ast} $-Algebras of Compact Operators

In this paper, we formulate and prove a version of the Stone-von Neumann Theorem for every $ C^{\ast} $-dynamical system of the form $ (G,\mathbb{K}(\mathcal{H}),α) $, where $ G $ is a locally compact Hausdorff abelian group and $ \mathcal{H} $ is a Hilbert space. The novelty of our work stems from our representation of the Weyl Commutation Relation on Hilbert $ \mathbb{K}(\mathcal{H}) $-modules instead of just Hilbert spaces, and our introduction of two additional commutation relations, which are necessary to obtain a uniqueness theorem. Along the way, we apply one of our basic results on Hil…

The Mazur-Ulam property for commutative von Neumann algebras

Let $(Ω,μ)$ be a $σ$-finite measure space. Given a Banach space $X$, let the symbol $S(X)$ stand for the unit sphere of $X$. We prove that the space $L^{\infty} (Ω,μ)$ of all complex-valued measurable essentially bounded functions equipped with the essential supremum norm, satisfies the Mazur-Ulam property, that is, if $X$ is any complex Banach space, every surjective isometry $Δ: S(L^{\infty} (Ω,μ))\to S(X)$ admits an extension to a surjective real linear isometry $T: L^{\infty} (Ω,μ)\to X$. This conclusion is derived from a more general statement which assures that every surjective isometry …

关于类型 $I$ Murray-von Neumann 代数中算子的 Jordan-Chevalley-Dunford 分解

On the Jordan-Chevalley-Dunford decomposition of operators in type $I$ Murray-von Neumann algebras

本文研究类型 $I$ Murray-von Neumann 代数中算子的 Jordan-Chevalley-Dunford 分解，证明矩阵分解映射的无界性，并给出稠密定义闭算子的典范分解，表明无界算子的必要性。

Affine density, von Neumann dimension and a problem of Perelomov

We provide a solution to Perelomov's 1972 problem concerning the existence of a phase transition (known in signal analysis as 'Nyquist rate') determining the basis properties of certain affine coherent states labelled by Fuchsian groups. As suggested by Perelomov, the transition is given according to the hyperbolic volume of the fundamental region. The solution is a more general form (in phase space) of the $PSL(2,\mathbb{R})$ variant of a 1989 conjecture of Kristian Seip about wavelet frames, where the same value of `Nyquist rate' is obtained as the trace of a certain localization operator. T…