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 $B(G)$.

For a closed subgroup of a locally compact group the Rieffel induction process gives rise to a C*-correspondence over the C*-algebra of the subgroup. We study the associated Cuntz–Pimsner algebra and show that, by varying the subgroup to be open, compact, or discrete, there are connections with the Exel–Pardo correspondence arising from a cocycle, and also with graph algebras.

We present a new construction of crossed-product duality for maximal coactions that uses Fischer’s work on maximalizations. Given a group $G$ and a coaction $(A,\unicode[STIX]{x1D6FF})$ we define a generalized fixed-point algebra as a certain subalgebra of $M(A\rtimes _{\unicode[STIX]{x1D6FF}}G\rtimes _{\,\widehat{\unicode[STIX]{x1D6FF}}}G)$, and recover the coaction via this double crossed product. Our goal is to formulate this duality in a category-theoretic context, and one advantage of our construction is that it breaks down into parts that are easy to handle in this regard. We first explain this for the category of nondegenerate *-homomorphisms and then, analogously, for the category of $C^{\ast }$-correspondences. Also, we outline partial results for the ‘outer’ category, which has been studied previously by the authors.

Proposition 2ċ5 of [5] states that a full coaction of a locally compact group on a C*-algebra is nondegenerate if and only if its normalisation is. Unfortunately, the proof there only addresses the forward implication, and we have not been able to find a proof of the opposite implication. This issue is important because the theory of crossed-product duality for coactions requires implicitly that the coactions involved be nondegenerate. Moreover, each type of coaction — full, reduced, normal, maximal, and (most recently) exotic — has its own distinctive properties with respect to duality, making it crucial to be able to convert from one to the other without losing nondegeneracy.

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).

We investigate topological realizations of higher-rank graphs. We show that the fundamental group of a higher-rank graph coincides with the fundamental group of its topological realization. We also show that topological realization of higher-rank graphs is a functor and that for each higher-rank graph Λ, this functor determines a category equivalence between the category of coverings of Λ and the category of coverings of its topological realization. We discuss how topological realization relates to two standard constructions for k-graphs: projective limits and crossed products by finitely generated free abelian groups.

In the third and latest paper in this series, we recover the imprimitivity theorems of Mansfield and Fell using our technique of Fell bundles over groupoids. Also, we apply the Rieffel surjection of the first paper in the series to relate our version of Mansfield’s theorem to that of an Huef and Raeburn, and to give an automatic amenability result for certain transformation Fell bundles.

For a countable discrete space $V$, every nondegenerate separable ${C}^{\ast } $-correspondence over ${c}_{0} (V)$ is isomorphic to one coming from a directed graph with vertex set $V$. In this paper we demonstrate why the analogous characterizations fail to hold for higher-rank graphs (where one considers product systems of ${C}^{\ast } $-correspondences) and for topological graphs (where $V$ is locally compact Hausdorff), and we discuss the obstructions that arise.

Cuntz and Li have defined a C*-algebra associated to any integral domain, using generators and relations, and proved that it is simple and purely infinite and that it is stably isomorphic to a crossed product of a commutative C*-algebra. We give an approach to a class of C*-algebras containing those studied by Cuntz and Li, using the general theory of C*-dynamical systems associated to certain semidirect product groups. Even for the special case of the Cuntz–Li algebras, our development is new.

We define the action of a locally compact group G on a topological graph E. This action induces a natural action of G on the C*-correspondence ℋ(E) and on the graph C*-algebra C*(E). If the action is free and proper, we prove that C*(E)⋊rG is strongly Morita equivalent to C*(E/G) . We define the skew product of a locally compact group G by a topological graph E via a cocycle c:E1 →G. The group acts freely and properly on this new topological graph E×cG. If G is abelian, there is a dual action on C* (E) such that . We also define the fundamental group and the universal covering of a topological graph.

Consider a projective limit G of finite groups Gn. Fix a compatible family δn of coactions of the Gn on a C*-algebra A. From this data we obtain a coaction δ of G on A. We show that the coaction crossed product of A by δ is isomorphic to a direct limit of the coaction crossed products of A by the δn. If A=C*(Λ) for some k-graph Λ, and if the coactions δn correspond to skew-products of Λ, then we can say more. We prove that the coaction crossed product of C*(Λ) by δ may be realized as a full corner of the C*-algebra of a (k+1)-graph. We then explore connections with Yeend’s topological higher-rank graphs and their C*-algebras.

We analyse Hecke pairs (G,H) and the associated Hecke algebra when G is a semi-direct product N ⋊ Q and H = M ⋊ R for subgroups M ⊂ N and R ⊂ Q with M normal in N. Our main result shows that, when (G,H) coincides with its Schlichting completion and R is normal in Q, the closure of in C*(G) is Morita–Rieffel equivalent to a crossed product I⋊βQ/R, where I is a certain ideal in the fixed-point algebra C*(N)R. Several concrete examples are given illustrating and applying our techniques, including some involving subgroups of GL(2,K) acting on K2, where K = ℚ or K = ℤ[p−1]. In particular we look at the ax + b group of a quadratic extension of K.

The Hecke algebra of a Hecke pair (G, H) is studied using the Schlichting completion (Ḡ, ), which is a Hecke pair whose Hecke algebra is isomorphic to and which is topologized so that is a compact open subgroup of Ḡ. In particular, the representation theory and C*-completions of are addressed in terms of the projection using both Fell's and Rieffel's imprimitivity theorems and the identity . An extended analysis of the case where H is contained in a normal subgroup of G (and in particular the case where G is a semi-direct product) is carried out, and several specific examples are analysed using this approach.

For a maximal coaction δ of a discrete group G on a C*-algebra A and a normal subgroup N of G, there are at least three natural A × G ×δ| N - A ×δ|1 G/N imprimitivity bimodules: Mansfield's bimodule ; the bimodule assembled by Ng from Green's imprimitivity bimodule and Katayama duality; and the bimodule assembled from and the crossed-product Mansfield bimodule . We show that all three of these are isomorphic, so that the corresponding inducing maps on representations are identical. This can be interpreted as saying that Mansfield and Green induction are inverses of one another ‘modulo Katayama duality’. These results pass to twisted coactions; dual results starting with an action are also given.

Let U, V, and W be multiplicative unitaries coming from discrete Kac systems such that W is an amenable normal submultiplicative unitary of V with quotient U. We define notions for right-Hilbert bimodules of coactions of SV and ŜV, their restrictions to SW and ŜU, their dual coactions, and their full and reduced crossed products. If N (A) denotes the imprimitivity bimodule associated to a coaction δ of SV on a C*-algebra A by Ng's imprimitivity theorem, we prove that for a suitably nondegenerate injective right-Hilbert bimodule coaction of SV on AXB, the balanced tensor products and are isomorphic right-Hilbert A×ŜV×rSU − B × ŜW bimodules. This can be interpreted as a natural equivalence between certain crossed-product functors.

We show that induction of covariant representations for C*-dynamical systems is natural in the sense that it gives a natural transformation between certain crossed-product functors. This involves setting up suitable categories of C*-algebras and dynamical systems, and extending the usual constructions of crossed products to define the appropriate functors. From this point of view, Green's Imprimitivity Theorem identifies the functors for which induction is a natural equivalence. Various special cases of these results have previously been obtained on an ad hoc basis.

Using the close relationship between coactions of discrete groups and Fell bundles, we introduce a procedure for inducing a ${{C}^{*}}$ -coaction $\delta :D\to D\otimes {{C}^{*}}\left( G/N \right)$ of a quotient group $G/N$ of a discrete group $G$ to a ${{C}^{*}}$-coaction $\text{Ind}\,\delta \text{:}\,\text{Ind}\,D\to \text{Ind}\,D\otimes {{C}^{*}}\left( G \right)\,\text{of }G$. We show that induced coactions behave in many respects similarly to induced actions. In particular, as an analogue of the well known imprimitivity theorem for induced actions we prove that the crossed products $\text{Ind}\,D{{\times }_{\text{Ind}\,\delta }}\,G\,\text{and }D{{\times }_{\delta }}\,G/N$ are always Morita equivalent. We also obtain nonabelian analogues of a theorem of Olesen and Pedersen which show that there is a duality between induced coactions and twisted actions in the sense of Green. We further investigate amenability of Fell bundles corresponding to induced coactions.

Groupoid actions on C*-bundles and inverse semigroup actions on C*-algebras are closely related when the groupoid is r-discrete.

We give a condition on a full coaction (A, G, δ) of a (possibly) non-amenable group G and a closed normal subgroup N of G which ensures that Mansfield imprimitivity works; i.e. that A×δ[mid ]G/N is Morita equivalent to A×δG×δˆrN. This condition obtains if N is amenable or δ is normal. It is preserved under Morita equivalence, inflation of coactions, the stabilization trick of Echterhoff and Raeburn, and on passing to twisted coactions.