Exotic coactions

TL;DR

This paper explores exotic coactions of locally compact groups on C*-algebras, focusing on those determined by ideals in the Fourier-Stieltjes algebra, and investigates their duality properties and classification.

Contribution

It introduces a framework for understanding exotic coactions via ideals in B(G), extending the theory of full and reduced crossed products.

Findings

Exotic coactions satisfy an intermediate duality called E-crossed product duality.

Not all coactions arise from ideals in B(G), but many do.

Partial classification of coactions determined by these ideals.

Abstract

If a locally compact group G acts on a C*-algebra B, we have both full and reduced crossed products, and each has a coaction of G. We investigate "exotic" coactions in between, that are determined by certain ideals E of the Fourier-Stieltjes algebra B(G) -- an approach that is inspired by recent work of Brown and Guentner on new C*-group algebra completions. We actually carry out the bulk of our investigation in the general context of coactions on a C*-algebra A. Buss and Echterhoff have shown that not every coaction comes from one of these ideals, but nevertheless the ideals do generate a wide array of exotic coactions. Coactions determined by these ideals E satisfy a certain "E-crossed product duality", intermediate between full and reduced duality. We give partial results concerning exotic coactions, with the ultimate goal being a classification of which coactions are determined by ideals of B(G).