Fourier--Stieltjes category for twisted groupoid actions
Alcides Buss, Bartosz Kwaśniewski, Andrew McKee, Adam Skalski
TL;DR
The paper extends Fourier–Stieltjes theory to twisted étale groupoid actions on arbitrary C*-bundles by building multiplier C*-correspondences, Fell bundles, and an accompanying dilation framework. A KSGNS–Murphy style dilation links positive-definite multipliers to completely positive fibre-preserving maps, while Fell absorption principles show that Fourier–Stieltjes multipliers act compatibly on reduced and full crossed products. The authors develop a rich categorical structure of Fourier–Stieltjes and Fourier multipliers, establish regular representations and absorption, and derive approximation properties with applications to weak containment and nuclearity. These results unify and generalize group-level harmonic analysis to the groupoid setting and suggest further directions for reconstruction, approximation properties, and extensions to Fell bundles.
Abstract
We extend the theory of Fourier--Stieltjes algebras to the category of twisted actions by étale groupoids on arbitrary C*-bundles, generalizing theories constructed previously by Bédos and Conti for twisted group actions on unital C*-algebras, and by Renault and others for groupoid C*-algebras, in each case motivated by the classical theory of Fourier--Stieltjes algebras of discrete groups. To this end we develop a toolbox including, among other things, a theory of multiplier C*-correspondences, multiplier C*-correspondence bundles, Busby--Smith twisted groupoid actions, and the associated crossed products, equivariant representations and Fell's absorption theorems. For a fixed étale groupoid $G$ a Fourier--Stieltjes multiplier is a family of maps acting on fibers, arising from an equivariant representation. It corresponds to a certain fiber-preserving strict completely bounded map between twisted full (or reduced) crossed products. We establish a KSGNS-type dilation result which shows that the correspondence above restricts to a bijection between positive-definite multipliers and a particular class of completely positive maps. Further we introduce a subclass of Fourier multipliers, that enjoys a natural absorption property with respect to Fourier--Stieltjes multipliers and gives rise to `reduced to full' multiplier maps on crossed products. Finally we provide several applications of the theory developed, for example to the approximation properties, such as weak containment or nuclearity, of the crossed products and actions in question, and discuss outstanding open problems.