A dynamical approach to sampling and interpolation in unimodular groups

Groupoids and Hermitian Banach *-algebras

“…In this section, we will show that groupoids arising from point sets as in e.g. [14] have polynomial growth under natural assumptions on the locally compact group and the point set. As such, they also lend themselves to applications of the results from Section 3 and Section 4.…”

“…2) The operations are the ones inherited from the transformation groupoid. It is shown in [14,Proposition 3.11] that when Λ is separated, G (Λ) is étale. Proposition 5.10.…”

A Groupoid Approach to Interacting Fermions

“…The mathematical aspects of patterns in non-abelian groups have been steadily attracted an interest from the mathematics community in the past few years [11,12,13,14,15,16,4]. In fact, during the completion of our work, Enstad, Raum and van Velthoven posted the works [27,28] on frame theory for patterns in locally compact second countable (lcsc) groups. These works not only cleared for us all the technical challenges, but they also supplied the natural mathematical framework we were looking for.…”

“…2) Every such closed set has a canonically associated lcsc étale groupoid G L and a separable C * -algebra C * r (G L ) [27].…”

“…The section also explains how the configurations of these physical systems are connected to patterns inside Iso(E d ) and it derives the specific form of the dynamical matrices under the assumption of Galilean invariance, which represents the point of departure for the operator theoretic framework. The first part of section 3 reproduces from [27] the relevant facts about the topological space of closed subsets of lcsc groups and specific classes of patterns. It also introduces the related concepts of hull and transveral of a pattern.…”