TENSOR-PRODUCT COACTION FUNCTORS

• Published in: Journal of the Australian Mathematical Society (2001) Vol. 112; no. 1; pp. 52 - 67
• Main Authors: KALISZEWSKI, S., LANDSTAD, MAGNUS B., QUIGG, JOHN
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.02.2022
• Subjects: Expanders; Mathematical analysis; Tensors; crossed product; coaction; Fell bundle; 46M15; action; 46L55; tensor product
• ISSN: 1446-7887; 1446-8107
• DOI: 10.1017/S1446788720000063

Recent work by Baum et al. [‘Expanders, exact crossed products, and the Baum–Connes conjecture’, Ann. K-Theory 1(2) (2016), 155–208], further developed by Buss et al. [‘Exotic crossed products and the Baum–Connes conjecture’, J. reine angew. Math. 740 (2018), 111–159], introduced a crossed-product functor that involves tensoring an action with a fixed action $(C,\unicode[STIX]{x1D6FE})$ , then forming the image inside the crossed product of the maximal-tensor-product action. For discrete groups, we give an analogue for coaction functors. We prove that composing our tensor-product coaction functor with the full crossed product of an action reproduces their tensor-crossed-product functor. We prove that every such tensor-product coaction functor is exact, and if $(C,\unicode[STIX]{x1D6FE})$ is the action by translation on $\ell ^{\infty }(G)$ , we prove that the associated tensor-product coaction functor is minimal, thereby recovering the analogous result by the above authors. Finally, we discuss the connection with the $E$ -ization functor we defined earlier, where $E$ is a large ideal of $B(G)$ .