#### Vol. 293, No. 2, 2018

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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
Primary: 46L55
Secondary: 46M15
##### Milestones
Revised: 1 September 2017
Accepted: 4 September 2017
Published: 23 November 2017
##### Authors
 S. Kaliszewski School of Mathematical and Statistical Sciences Arizona State University Tempe, AZ 85287-1804 United States Magnus B. Landstad Department of Mathematical Sciences Norwegian University of Science and Technology 7491 Trondheim Norway John Quigg School of Mathematical and Statistical Sciences Arizona State University Tempe, AZ 85287-1804 United States