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We prove a version of Pedersen's outer conjugacy theorem for coactions of compact groups, which characterizes outer conjugate coactions of a compact group in terms of properties of the dual actions. In fact, we show that every isomorphism of a dual action comes from a unique outer conjugacy of a coaction, which in this context should be called strong Pedersen rigidity. We promote this to a category equivalence. [ABSTRACT FROM AUTHOR]

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 $B(G)$. [ABSTRACT FROM AUTHOR]

Given a group cocycle on a finitely aligned left cancellative small category (LCSC), we investigate the associated skew product category and its Cuntz-Krieger algebra, which we describe as the crossed product of the Cuntz-Krieger algebra of the original category by an induced coaction of the group. We use our results to study Cuntz-Krieger algebras arising from free actions of groups on finitely aligned LCSCs, and to construct coactions of groups on Exel-Pardo algebras. Finally, we discuss the universal group of a small category and connectedness of skew product categories. [ABSTRACT FROM AUTHOR]

This is a follow-up to a paper with the same title and by the same authors. In that paper, all groups were assumed to be abelian, and we are now aiming to generalize the results to nonabelian groups. The motivating point is Pedersen's theorem, which does hold for an arbitrary locally compact group G , saying that two actions (A , α) and (B , β) of G are outer conjugate if and only if the dual coactions (A ⋊ α G , α ̂) and (B ⋊ β G , β ̂) of G are conjugate via an isomorphism that maps the image of A onto the image of B (inside the multiplier algebras of the respective crossed products). We do not know of any examples of a pair of non-outer-conjugate actions such that their dual coactions are conjugate, and our interest is therefore exploring the necessity of latter condition involving the images; and we have decided to use the term "Pedersen rigid" for cases where this condition is indeed redundant. There is also a related problem, concerning the possibility of a so-called equivariant coaction having a unique generalized fixed-point algebra, that we call "fixed-point rigidity". In particular, if the dual coaction of an action is fixed-point rigid, then the action itself is Pedersen rigid, and no example of non-fixed-point-rigid coaction is known. [ABSTRACT FROM AUTHOR]

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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. [ABSTRACT FROM AUTHOR]

Let be a locally compact abelian group. By modifying a theorem of Pedersen, it follows that actions of on -algebras and are outer conjugate if and only if there is an isomorphism of the crossed products that is equivariant for the dual actions and preserves the images of and in the multiplier algebras of the crossed products. The rigidity problem discussed in this paper deals with the necessity of the last condition concerning the images of and . There is an alternative formulation of the problem: an action of the dual group together with a suitably equivariant unitary homomorphism of give rise to a generalized fixed-point algebra via Landstad's theorem, and a problem related to the above is to produce an action of and two such equivariant unitary homomorphisms of that give distinct generalized fixed-point algebras. We present several situations where the condition on the images of and is redundant, and where having distinct generalized fixed-point algebras is impossible. For example, if is discrete, this will be the case for all actions of . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We prove that if E and F are large ideals of B(G) for which the associated coaction functors are exact, then the same is true for E ∩ F. We also give an example of a coaction functor whose restriction to the maximal coactions does not come from any large ideal. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

In this partly expository paper we compare three different categories of C*-algebras in which crossed-product duality can be formulated, both for actions and for coactions of locally compact groups. In these categories, the isomorphisms correspond to C*-algebra isomorphisms, imprimitivity bimodules, and outer conjugacies, respectively. In each case, a variation of the fixed-point functor that arises from classical Landstad duality is used to obtain a quasi-inverse for a crossed-product functor. To compare the various cases, we describe in a formal way our view of the fixed-point functor as an "inversion" of the process of forming a crossed product. In some cases, we obtain what we call "good" inversions, while in others we do not. For the outer-conjugacy categories, we generalize a theorem of Pedersen to obtain a fixed-point functor that is quasi-inverse to the reduced-crossed-product functor for actions, and we show that this gives a good inversion. For coactions, we prove a partial version of Pedersen's theorem that allows us to define a fixed-point functor, but the question of whether it is a quasi-inverse for the crossed-product functor remains open. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

Abstract: We show that the passage from a -correspondence to its Cuntz–Pimsner -algebra gives a functor on a category of -correspondences with appropriately defined morphisms. Applications involving topological graph -algebras are discussed, and an application to crossed-product correspondences is presented in detail. [Copyright &y& Elsevier]

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. [ABSTRACT FROM PUBLISHER]

We show that for a locally compact group G there is a one-to-one correspondence between G-invariant weak*-closed subspaces E of the Fourier-Stieltjes algebra B(G) containing Br(G) and quotients C*E (G) of C*(G) which are intermediate between C*(G) and the reduced group algebra C*r (G). We show that the canonical comultiplication on C*(G) descends to a coaction or a comultiplication on C*E (G) if and only if E is an ideal or subalgebra, respectively. When α is an action of G on a C*-algebra B, we define "E-crossed products" B ⋊α,E G lying between the full crossed product and the reduced one, and we conjecture that these intermediate crossed products "satisfy" an "exotic" version of crossed-product duality involving C*E(G). [ABSTRACT FROM AUTHOR]

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 $C^*(E)\rtimes \hat {G}\cong C^*(E\times _cG)$. We also define the fundamental group and the universal covering of a topological graph. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 eδ 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 D 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. [ABSTRACT FROM AUTHOR]

Abstract: Suppose a locally compact group G acts freely and properly on a locally compact Hausdorff space X, and let γ be the induced action on . We consider a category in which the objects are -dynamical systems for which there is an equivariant homomorphism of into the multiplier algebra . Rieffel has shown that such systems are proper and saturated, and hence have a generalized fixed-point algebra which is Morita equivalent to . We show that the assignment is functorial, and that Rieffel''s Morita equivalence is natural in a suitable sense. We then use our results to prove a categorical version of Landstad duality which characterizes crossed products by coactions, and to prove that Mansfield imprimitivity for crossed products by homogeneous spaces is natural. [Copyright &y& Elsevier]