Coaction functors, II

S. Kaliszewski, Magnus B. Landstad and John Quigg

Vol. 293 (2018), No. 2, 301–339

DOI: 10.2140/pjm.2018.293.301

Abstract

In their study of the application of crossed-product functors to the Baum–Connes conjecture, Buss, Echterhoff, and Willett introduced various properties that crossed-product functors may have. Here we introduce and study analogues of some of these properties for coaction functors, making sure that the properties are preserved when the coaction functors are composed with the full crossed product to make a crossed-product functor. The new properties for coaction functors studied here are functoriality for generalized homomorphisms and the correspondence property. We also study the connections with the ideal property. The study of functoriality for generalized homomorphisms requires a detailed development of the Fischer construction of maximalization of coactions with regard to possibly degenerate homomorphisms into multiplier algebras. We verify that all “KLQ” functors arising from large ideals of the Fourier–Stieltjes algebra $B\left(G\right)$ have all the properties we study, and at the opposite extreme we give an example of a coaction functor having none of the properties.

Keywords

crossed product, action, coaction, Fourier–Stieltjes algebra, exact sequence, Morita compatible

Mathematical Subject Classification 2010

Primary: 46L55

Secondary: 46M15

Milestones

Received: 9 January 2017

Revised: 1 September 2017

Accepted: 4 September 2017

Published: 23 November 2017