Fourier--Stieltjes category for twisted groupoid actions

TL;DR

This paper generalizes Fourier--Stieltjes algebras to twisted actions of étale groupoids on C*-bundles, developing a comprehensive framework with new tools and applications to approximation properties of crossed products.

Contribution

It introduces a new theory of Fourier--Stieltjes multipliers for twisted groupoid actions, extending previous work and establishing a KSGNS-type dilation theorem.

Findings

Established a bijection between positive-definite multipliers and completely positive maps.

Developed a framework for multiplier C*-correspondences and their bundles.

Applied the theory to approximation properties like nuclearity and weak containment.

Abstract

We extend the theory of Fourier--Stieltjes algebras to the category of twisted actions by \'etale groupoids on arbitrary C*-bundles, generalizing theories constructed previously by B\'{e}dos 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 \'etale 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.