# Coaction functors, II

## TL;DR

This paper extends the theory of coaction functors by introducing and analyzing properties like functoriality and the correspondence property, with implications for the Baum-Connes Conjecture.

## Contribution

It develops analogues of properties for coaction functors, ensuring their preservation under composition with full crossed products, and examines their connections with the ideal property.

## Key findings

- KLQ functors from large ideals of B(G) have all studied properties.
- An example of a coaction functor lacking all properties is provided.
- The study enhances understanding of coaction functors in the context of crossed-product functors.

## Abstract

In further study of the application of crossed-product functors to the Baum-Connes Conjecture, Buss, Echterhoff, and Willett introduced various other properties that crossed-product functors may have. Here we introduce and study analogues 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 particularly 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 $B(G)$ have all the properties we study, and at the opposite extreme we give an example of a coaction functor having none of the properties.