Coactions and skew products for topological graphs
The C*-algebra of a skew-product topological graph is a crossed product of the C*-algebra of the base topological graph by a coaction.
math.OA↗
arXiv subjects
S. Kaliszewski
Publications and source records attributed to S. Kaliszewski.
Coactions on Cuntz-Pimsner algebras
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.
math.OA↗
Fell bundles and imprimitivity theorems: towards a universal generalized fixed point algebra
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.
math.OA↗
Categorical perspectives in noncommutative duality
Noncommutative duality for C*-dynamical systems is a vast generalization of Pontryagin duality for locally compact abelian groups. In this series of lectures, we give an introduction to the categorical aspects of this duality, focusing primarily on Landstad duality for actions and coactions of locally compact groups.
math.OA↗
Functoriality of Cuntz-Pimsner correspondence maps
We show that the passage from a $C^\ast$-correspondence to its Cuntz-Pimsner $C^\ast$-algebra gives a functor on a category of $C^\ast$-correspondences with appropriately defined morphisms. Applications involving topological graph $C^\ast$-algebras are discussed, and an application to crossed-product correspondences is presented in detail.
math.OA↗
Fell bundles and imprimitivity theorems
Our goal in this paper and two sequels is to apply the Yamagami-Muhly-Williams equivalence theorem for Fell bundles over groupoids to recover and extend all known imprimitivity theorems involving groups. Here we extend Raeburn's symmetric imprimitivity theorem, and also, in an appendix, we develop a number of tools for the theory of Fell bundles that have not previously appeared in the literature.
math.OA↗
Obstructions to a general characterization of graph correspondences
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.
math.OA↗
Inner coactions, Fell bundles, and abstract uniqueness theorems
We prove gauge-invariant uniqueness theorems with respect to maximal and normal coactions for $C^*$-algebras associated to product systems of $C^*$-correspondences. Our techniques of proof are developed in the abstract context of Fell bundles. We employ inner coactions to prove an essential-inner uniqueness theorem for Fell bundles. As application, we characterise injectivity of homomorphisms on Nica's Toeplitz algebra $\Tt(G, P)$ of a quasi-lattice ordered group $(G, P)$ in the presence of a finite non-trivial set of lower bounds for all non-trivial elements in $P$.
math.OA↗
Topological realizations and fundamental groups of higher-rank graphs
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.
math.CO↗
On reflective-coreflective equivalence and associated pairs
We show that a reflective/coreflective pair of full subcategories satisfies a "maximal-normal"-type equivalence if and only if it is an associated pair in the sense of Kelly and Lawvere.
math.OA↗
A crossed-product approach to the Cuntz-Li algebras
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.
math.OA↗
Naturality of Symmetric Imprimitivity Theorems
The first imprimitivity theorems identified the representations of groups or dynamical systems which are induced from representations of a subgroup. Symmetric imprimitivity theorems identify pairs of crossed products by different groups which are Morita equivalent, and hence have the same representation theory. Here we consider commuting actions of groups $H$ and $K$ on a $C^*$-algebra which are saturated and proper as defined by Rieffel in 1990. Our main result says that the resulting Morita equivalence of crossed products is natural in the sense that it is compatible with homomorphisms and induction processes.
math.OA↗
Reflective-coreflective equivalence
We explore a curious type of equivalence between certain pairs of reflective and coreflective subcategories. We illustrate with examples involving noncommutative duality for C*-dynamical systems and compact quantum groups, as well as examples where the subcategories are actually isomorphic.
math.OA↗
Characterizing graph C*-correspondences
Every separable nondegenerate C*-correspondence over a commutative C*-algebra with discrete spectrum is isomorphic to a graph correspondence.
math.OA↗
Coactions and Fell bundles
We show that if $Å$ is a Fell bundle over a locally compact group $G$, then there is a natural coaction $δ$ of $G$ on the Fell-bundle $C^*$-algebra $C^*(G,Å)$ such that if $\hatδ$ is the dual action of $G$ on the crossed product $C^*(G,Å) \rtimes_δ G$, then the full crossed product $(C^*(G,Å) \rtimes_δG)\rtimes_{\hatδ}G$ is canonically isomorphic to $C^*(G,Å) \otimes\KK(L^2(G))$. Hence the coaction $δ$ is maximal.
math.OA↗
Fixed-point algebras for proper actions and crossed products by homogeneous spaces
We consider a fixed free and proper action of a locally compact group $G$ on a space $T$, and actions $α:G\to \Aut A$ on $C^*$-algebras for which there is an equivariant embedding of $(C_0(T),\rt)$ in $(M(A),α)$. A recent theorem of Rieffel implies that $α$ is proper and saturated with respect to the subalgebra $C_0(T)AC_0(T)$ of $A$, so that his general theory of proper actions gives a Morita equivalence between $A\rtimes_{α,r}G$ and a generalised fixed-point algebra $A^α$. Here we investigate the functor $(A,α)\mapsto A^α$ and the naturality of Rieffel's Morita equivalence, focusing in particular on the relationship between the different functors associated to subgroups and quotients. We then use the results to study induced representations for crossed products by coactions of homogeneous spaces $G/H$ of $G$, which were previously shown by an Huef and Raeburn to be fixed-point algebras for the dual action of $H$ on the crossed product by $G$.
math.OA↗
Naturality of Rieffel's Morita equivalence for proper actions
Suppose that a locally compact group $G$ acts freely and properly on the right of a locally compact space $T$. Rieffel proved that if $α$ is an action of $G$ on a $C^*$-algebra $A$ and there is an equivariant embedding of $C_0(T)$ in $M(A)$, then the action $α$ of $G$ on $A$ is proper, and the crossed product $A\rtimes_{α,r}G$ is Morita equivalent to a generalised fixed-point algebra $\Fix(A,α)$ in $M(A)^α$. We show that the assignment $(A,α)\mapsto\Fix(A,α)$ extends to a functor $\Fix$ on a category of $C^*$-dynamical systems in which the isomorphisms are Morita equivalences, and that Rieffel's Morita equivalence implements a natural isomorphism between a crossed-product functor and $\Fix$. From this, we deduce naturality of Mansfield imprimitivity for crossed products by coactions, improving results of Echterhoff-Kaliszewski-Quigg-Raeburn and Kaliszewski-Quigg Raeburn, and naturality of a Morita equivalence for graph algebras due to Kumjian and Pask.
math.OA↗
Proper actions, fixed-point algebras and naturality in nonabelian duality
Suppose a locally compact group G acts freely and properly on a locally compact Hausdorff space X, and let gamma be the induced action on C_0(X). We consider a category in which the objects are C*-dynamical systems (A, G, alpha) for which there is an equivariant homomorphism of (C_0(X), gamma) into the multiplier algebra M(A). Rieffel has shown that such systems are proper and saturated, and hence have a generalized fixed-point algebra A^alpha which is Morita equivalent to A times_{alpha,r} G. We show that the assignment (A, alpha) maps to A^alpha 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.
math.OA↗