Exotic Coactions

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 the two, which 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).

Coaction functors

“…is[KLQ, Theorem 4.6], and (2) trivially implies (3). Assume (3), i.e., that (B, ε) is maximal and we have an isomorphism θ :(B E , ε E ) → (A, δ).…”

“…The most important source of examples of the decreasing coaction functors of the preceding section is large ideals. We recall some basic concepts from [KLQ13,KLQ]. Let E be an ideal of B(G) that is large, meaning it is nonzero, G-invariant, and weak* closed.…”

“…Although E-ization and τ E have similar properties, they are not naturally isomorphic functors (see Remark 6.15). The outputs of E-ization are precisely the coactions we call E-coactions, namely those for which E-crossed-product duality holds [KLQ,Theorem 4.6] (see also [BE13, Theorem 5.1]). Theorem 6.17 shows that τ E gives an equivalence of maximal coactions with E-coactions.…”

Operator Algebras and Their Applications

“…In [KLQ,Definition 5.1] we introduced the above properness conditions, but in that paper we used the term proper coaction for the above sproper coaction, and slice proper coaction for the above w-proper coaction (because it involves the slice map ! ⌦ id).…”

“…In [KLQ,Theorem 6.10] we used this Galois correspondence to exhibit examples of small ideals J that are not of the form J (E) for any large ideal E. Buss and Echterho↵ [BE13,Example 5.3] have given examples that are better in the sense that the coaction (A, ) is of the form (B o ↵ G, b ↵). Consequently, there are exotic crossed products that are not E-crossed products for any large ideal E.…”