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For a discrete group $G$, we develop a `$G$-balanced tensor product' of two coactions $(A,\delta)$ and $(B,\epsilon)$, which takes place on a certain subalgebra of the maximal tensor product $A\otimes_{\max} B$. Our motivation for this is that we are able to prove that given two actions of $G$, the dual coaction on the crossed product of the maximal-tensor-product action is isomorphic to the $G$-balanced tensor product of the dual coactions. In turn, our motivation for this is to give an analogue, for coaction functors, of a crossed-product functor originated by Baum, Guentner, and Willett, and further developed by Buss, Echterhoff, and Willett, that involves tensoring an action with a fixed action $(C,\gamma)$, then forming the image inside the crossed product of the maximal-tensor-product action. We prove that composing our tensor-product coaction functor with the full crossed product of an action reproduces the tensor-crossed-product functor of Baum, Guentner, and Willett. We prove that every such tensor-product coaction functor is exact, thereby recovering the analogous result for the tensor-crossed-product functors of Baum, Guentner, and Willett. When $(C,\gamma)$ is the action by translation on $\ell^\infty(G)$, we prove that the associated tensor-product coaction functor is minimal, generalizing the analogous result of Buss, Echterhoff, and Willett for tensor-crossed-product functors., Comment: Minor revision

espanolSi α es una accion de un grupo abeliano localmente compacto G sobre una C*-algebra A, la dualidad de Takesaki-Takai recupera (A, α), salvo equivalencia de Morita, de la accion dual de Ĝ sobre el producto cruzado A × α G. Mediante un poco mas de informacion, la dualidad de Landstad recupera (A, α) salvo isomorfismo. De manera intermedia, mediante la modificacion de un teorema de Pedersen, (A α) es recuperado, salvo conjugacion externa, de la accion dual y de la posicion de A en M(A ×α G). Nuestra busqueda (todavia sin exito, de alguna manera irritante) de ejemplos que prueben la necesidad de esta ultima condicion, nos ha conducido a a formular el "problema de rigidez de Pedersen". Presentamos numerosas situaciones donde la condicion es redundante, incluidos los casos en que G es discreto, o bien A es estable o conmutativo. Lo mas interesante de estos "teoremas de no usar" es para acciones localmente unitarias sobre algebras trazo-continuas. EnglishIf is an action of a locally compact abelian group G on a C*-algebra A, Takesaki-Takai duality recovers (A, α) up to Morita equivalence from the dual action of Ĝ on the crossed product A × α G. Given a bit more information, Landstad duality recovers (A, α) up to isomorphism. In between these, by modifying a theorem of Pedersen, (A, α) is recovered up to outer conjugacy from the dual action and the position of A in M(A ×α G). Our search (still unsuccessful, somehow irritating) for examples showing the necessity of this latter condition has led us to formulate the "Pedersen Rigidity problem". We present numerous situations where the condition is redundant, including G discrete or A stable or commutative. The most interesting of these "no-go theorems" is for locally unitary actions on continuous-trace algebras.

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 $(A,\alpha)$ and $(B,\beta)$ of $G$ are outer conjugate if and only if the dual coactions $(A\rtimes_{\alpha}G,\widehat\alpha)$ and $(B\rtimes_{\beta}G,\widehat\beta)$ 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., Comment: Minor revision. To appear in Internat. J. Math

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 $(A,\delta)$ we define a generalized fixed-point algebra as a certain subalgebra of $M(A\rtimes_{\delta} G \rtimes_{\widehat{\delta}} G)$, 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 $C^*$-correspondences. Also, we outline partial results for the "outer" category, studied previously by the authors., Comment: Minor revision

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, that 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)., corrected and shortened

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., minor revisions

S. KALISZEWSKI, PAUL S. MUHLY, JOHN QUIGG, AND DANA P. WILLIAMSAbstract. In the third and latest paper in this series, we recover the imprim-itivity theorems of Mansfield and Fell using our technique of Fell bundles overgroupoids. Also, we apply the Rieffel Surjection of the first paper in the seriesto relate our version of Mansfield’s theorem to that of an Huef and Raeburn,and to give an automatic amenability result for certain transformation Fellbundles.

For a countable discrete space V, every nondegenerate separable C*-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*-correspondences) and for topological graphs (where V is locally compact Hausdorff), and we discuss the obstructions that arise., Comment: major revision; stated some results in greater generality

Cuntz and Li have defined aC*-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 commutativeC*-algebra. We give an approach to a class ofC*-algebras containing those studied by Cuntz and Li, using the general theory ofC*-dynamical systems associated to certain semidirect product groups. Even for the special case of the Cuntz–Li algebras, our development is new.

We survey the results required to pass between full and reduced coactions of locally compact groups on C ∗ -algebras, which say, roughly speaking, that one can always do so without changing the crossed-product C ∗ -algebra. Wherever possible we use definitions and constructions that are well documented and accessible to non-experts, and otherwise we provide full details. We then give a series of applications to illustrate the use of these techniques. We obtain in particular a new version of Mansfield’s imprimitivity theorem for full coactions, and prove that it gives a natural isomorphism between crossed-product functors defined on appropriate categories.

Let $G$ be a locally compact abelian group. By modifying a theorem of Pedersen, it follows that actions of $G$ on $C^*$-algebras $A$ and $B$ 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 $A$ and $B$ 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 $A$ and $B$. There is an alternative formulation of the problem: an action of the dual group $\hat G$ together with a suitably equivariant unitary homomorphism of $G$ 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 $\hat G$ and two such equivariant unitary homomorphisms of $G$ that give distinct generalized fixed-point algebras. We present several situations where the condition on the images of $A$ and $B$ is redundant, and where having distinct generalized fixed-point algebras is impossible. For example, if $G$ is discrete, this will be the case for all actions of $G$., Comment: minor revision

We analyse Hecke pairs (G,H) and the associated Hecke algebra $\mathcal{H}$ 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 $\mathcal{H}$ 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.

We show that the category A(G) of actions of a locally compact group G on C*-algebras (with equivariant nondegenerate *-homomorphisms into multiplier algebras) is equivalent, via a full-crossed-product functor, to a comma category of maximal coactions of G under the comultiplication (C*(G),delta_G); and also that A(G) is equivalent, via a reduced-crossed-product functor, to a comma category of normal coactions under the comultiplication. This extends classical Landstad duality to a category equivalence, and allows us to identify those C*-algebras which are isomorphic to crossed products by G as precisely those which form part of an object in the appropriate comma category., 29 pages. Extensively revised, although essentially the same main results

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.

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 the intersection of E and F. We also give an example of a coaction functor whose restriction to the maximal coactions does not come from any large ideal., minor revision

A certain type of functor on a category of coactions of a locally compact group on C*-algebras is introduced and studied. These functors are intended to help in the study of the crossed-product functors that have been recently introduced in relation to the Baum-Connes conjecture. The most important coaction functors are the ones induced by large ideals of the Fourier-Stieltjes algebra. It is left as an open problem whether the "minimal exact and Morita compatible crossed-product functor" is induced by a large ideal., Comment: Minor corrections, and fixed proof of Proposition 4.24

For any maximal coaction ( A , G , δ ) (A,G,\delta ) and any closed normal subgroup N N of G G , there exists an imprimitivity bimodule Y G / N G ( A ) Y_{G/N}^G(A) between the full crossed product A × δ G × δ ^ | N A\times _\delta G\times _{\widehat \delta |}N and A × δ | G / N A\times _{\delta |}G/N , together with Inf ⁡ δ ^ | ^ − δ dec \operatorname {Inf}\widehat {\widehat \delta |}-\delta ^{\text {dec}} compatible coaction δ Y \delta _Y of G G . The assignment ( A , δ ) ↦ ( Y G / N G ( A ) , δ Y ) (A,\delta )\mapsto (Y_{G/N}^G(A),\delta _Y) implements a natural equivalence between the crossed-product functors “ × G × N {}\times G\times N ” and “ × G / N {}\times G/N ”, in the category whose objects are maximal coactions of G G and whose morphisms are isomorphism classes of right-Hilbert bimodule coactions of G G .

This short note amends Proposition 2.5 of the named article, which states that a full coaction of a locally compact group on a C*-algebra is nondegenerate if and only if its normalization is. The proof given there of the reverse implication is incorrect and, unfortunately, we have been unable to find a correct proof. Instead, we summarize the current state of our knowledge of the relationships between nondegeneracy of normal, reduced, and generic C*-coactions., Removed stuff on maximal coactions

We will define new constructions similar to the graph systems of correspondences described by Deaconu et al. We will use these to prove a version of Ionescu's theorem for higher rank graphs. Afterwards we will examine the properties of these constructions further and make contact with Yeend's topological k-graphs and the tensor groupoid valued product systems of Fowler and Sims., Comment: minor changes

There are at least three imprimitivity bimodules naturally associated to a maximal coaction of a discrete group G on a C*-algebra and a normal subgroup of G: Mansfield's bimodule; the bimodule assembled by Ng from Green's imprimitivity bimodule and Katayama duality; and a bimodule assembled from Green's bimodule and a 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., Comment: LaTeX-2e, 20 pages, uses packages amssymb, xy, upref

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.

Using the close relationship between coactions of discrete groups and Fell bundles, we introduce a procedure for inducing a C*-coaction δ: D → D ⊗C*(G/N) of a quotient group G/N of a discrete group G to a C*-coaction Ind δ: Ind D → D ⊗C*(G) 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 Ind D ×IndδG and D ×δG/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.

We give a short proof of a recent theorem of Ionescu which shows that the Cuntz-PimsnerC∗C^*-algebra of a certain correspondence associated to a Mauldin-Williams graph is isomorphic to the graph algebra.

Suppose N N is a closed normal subgroup of a locally compact group G G . A coaction : A → M ( A ⊗ C ∗ ( N ) ) :A \to M(A \otimes {C^ * }(N)) of N N on a C ∗ {C^ * } -algebra A A can be inflated to a coaction δ \delta of G G on A A , and the crossed product A × δ G A{ \times _\delta }G is then isomorphic to the induced C ∗ {C^ * } -algebra Ind N G A × ϵ N \text {Ind}_N^G A{\times _\epsilon }N . We prove this and a natural generalization in which A × ϵ N A{ \times _\epsilon }N is replaced by a twisted crossed product A × G / N G A{ \times _{G/N}}G ; in case G G is abelian, we recover a theorem of Olesen and Pedersen. We then use this to extend the Landstad duality of the first author to twisted crossed products, and give several applications. In particular, we prove that if \[ 1 → N → G → G / N → 1 1 \to N \to G \to G/N \to 1 \] is topologically trivial, but not necessarily split as a group extension, then every twisted crossed product A × G / N G A{ \times _{G/N}}G is isomorphic to a crossed product of the form A × N A \times N .

We apply the One-Sided Action Theorem from the first paper in this series to prove that Rieffel's Morita equivalence between the reduced crossed product by a proper saturated action and the generalized fixed-point algebra is a quotient of a Morita equivalence between the full crossed product and a "universal" fixed-point algebra. We give several applications, to Fell bundles over groups, reduced crossed products as fixed-point algebras, and C*-bundles., minor revision

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 \Lambda, this functor determines a category equivalence between the category of coverings of \Lambda\ 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., Comment: 23 pages; pictures prepared with TiKZ

Full and reduced C*-coactions are shown to be essentially equivalent as far as the representations and cocrossed products are concerned, at least in the presence of non-degeneracy. This is shown to be particularly true for a special class of full coactions which are given the name normal.

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 ${\mathcal H}(E)$ and on the graph $C^*$-algebra $C^*(E)$. If the action is free and proper, we prove that $C^*(E)\rtimes_r G$ 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 $c:E^1\to G$. The group acts freely and properly on this new topological graph $E\times_cG$. 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., We corrected a gap in the proof of Thm 5.6

Consider a projective limit G of finite groups G_n. Fix a compatible family \delta^n of coactions of the G_n on a C*-algebra A. From this data we obtain a coaction \delta of G on A. We show that the coaction crossed product of A by \delta is isomorphic to a direct limit of the coaction crossed products of A by the \delta^n. If A = C*(\Lambda) for some k-graph \Lambda, and if the coactions \delta^n correspond to skew-products of \Lambda, then we can say more. We prove that the coaction crossed-product of C*(\Lambda) by \delta may be realised 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., Comment: 19 pages, laTeX. v2: Minor modifications to version 1. This version to appear in the Journal of the Australian Mathematical Society v3: some potentially confusing typos corrected in the proof of Theorem~3.1, as well as a few others. References updated

We investigate how a correspondence coaction gives rise to a coaction on the associated Cuntz-Pimsner algebra. We apply this to recover a recent result of Hao and Ng concerning Cuntz-Pimsner algebras of crossed products of correspondences by actions of amenable groups., 25 pages

Introduction Right-Hilbert bimodules The categories The functors The natural equivalences Applications Appendix A. Crossed products by actions and coactions Appendix B. The imprimitivity theorems of Green and Mansfield Appendix C. function spaces Appendix. Bibliography.

A coaction d of a locally compact group G on a C*-algebra A is maximal if a certain natural map from A times_d G times_{d hat} G onto A otimes K(L^2(G)) is an isomorphism. All dual coactions on full crossed products by group actions are maximal; a discrete coaction is maximal if and only if A is the full cross-sectional algebra of the corresponding Fell bundle. For every nondegenerate coaction of G on A, there is a maximal coaction of G on an extension of A such that the quotient map induces an isomorphism of the crossed products., Comment: 12 pages

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 spcecial cases of these results have previously been obtained on an ad hoc basis., LaTeX-2e, 24 pages, uses package pb-diagram

We give a condition on a full coaction $(A,G,\delta)$ of a (possibly) nonamenable group $G$ and a closed normal subgroup $N$ of $G$ which ensures that Mansfield imprimitivity works; i.e. that $A\times_{\delta{\vert}} G/N$ is Morita equivalent to $A\times_\delta G\times_{\deltahat,r} N$. This condition obtains if $N$ is amenable or $\delta$ is normal. It is preserved under Morita equivalence, inflation of coactions, the stabilization trick of Echterhoff and Raeburn, and on passing to twisted coactions., Comment: 23 pages, LaTeX 2e, requires amscd.sty and pb-diagram.sty. Revisions include deletion of false Lemma 2.3 and amendment of proofs of Proposition 2.4 and Theorem 4.1, which had relied on the false lemma or its proof

For partial automorphisms of $C^*$-algebras, Takai-Takesaki crossed product duality tends to fail, in proportion to the extent to which the partial automorphism is not an automorphism., Comment: 18 pages, AMS-LaTeX

Using the strong relation between coactions of a discrete group G on C*-algebras and Fell bundles over G, we prove a new version of Mansfield's imprimitivity theorem for coactions of discrete groups. Our imprimitivity theorem works for the universally defined full crossed products and arbitrary subgroups of G, as opposed to the usual theory which uses the spatially defined reduced crossed products and normal subgroups of G. Moreover, our theorem factors through the usual one by passing to appropriate quotients. As applications we show that a Fell bundle over a discrete group is amenable in the sense of Exel if and only if the double dual action is amenable in the sense that the maximal and reduced crossed products coincide. We also give a new characterization of induced coactions in terms of their dual actions., Comment: 18 pages

A unitary representation induced from a normal subgroup of a separable locally compact group is irreducible if and only if the inducing representation is irreducible and the restriction of the induced representation to the normal subgroup is multiplicity-free.

In the duality for locally compact groups, much use is made of a version of the Hopf algebra technique in the context of von Neumann algebras, culminating in the theory of Kac algebras [6], [14]. It seems natural to ask whether something like a Hopf algebraic structure can be defined on the pre-dual of a Kac algebra. This leads to the question of whether the multiplication on a von Neumann algebra M, viewed as a linear map m from M ⊙ M (the algebraic tensor product) to M, can be pre-transposed to give a co-multiplication on the pre-dual M*, i.e., a linear map m* from M* to the completion of M* ⊙ M* with respect to some cross-norm. A related question is whether the multiplication on a C*-algebra A can be transposed to give a co-multiplication on the dual A*. Of course, this can be regarded as a special case of the preceding question by taking M = A**, where the double dual A** is identified with the enveloping von Neumann algebra of A.

A unitary representation induced from a normal subgroup of a separable locally compact group is irreducible if and only if the inducing representation is irreducible and the restriction of the induced representation to the normal subgroup is multiplicity-free.

Huruya [4] has proven that, for C ∗ {C^*} -algebras A 1 {A_1} and A 2 {A_2} , \[ ( A 1 ⊗ A 2 ) ∗ ∗ = A 1 ∗ ∗ ⊗ ¯ A 2 ∗ ∗ {({A_1} \otimes {A_2})^{**}} = A_1^{**}\overline \otimes A_2^{**} \] for every A 2 {A_2} if and only if A 1 {A_1} is scattered. We strengthen this by proving that ( A 1 ⊗ A 2 ) ∗ ∗ = A 1 ∗ ∗ ⊗ ¯ A 2 ∗ ∗ {({A_1} \otimes {A_2})^{**}} = A_1^{**}\overline \otimes A_2^{**} if and only if A 1 {A_1} or A 2 {A_2} is scattered. We discuss ramifications to representation theory and related questions regarding normal representations of W ∗ {W^*} -tensor products.