Results 1  10
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15
Equivariant Kasparov theory and generalized homomorphisms, Ktheory 21
, 2000
"... Abstract. Let G be a locally compact group. We describe elements of KKG (A, B) by equivariant homomorphisms, following Cuntz’s treatment in the nonequivariant case. This yields another proof for the universal property of KKG: It is the universal split exact stable homotopy functor. To describe a Ka ..."
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Cited by 22 (7 self)
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Abstract. Let G be a locally compact group. We describe elements of KKG (A, B) by equivariant homomorphisms, following Cuntz’s treatment in the nonequivariant case. This yields another proof for the universal property of KKG: It is the universal split exact stable homotopy functor. To describe a Kasparov triple (E, φ, F) for A, B by an equivariant homomorphism, we have to arrange for the Fredholm operator F to be equivariant. This can be done if A is of the form K(L2G)⊗A ′ and more generally if the group action on A is proper in the sense of Exel and Rieffel. 1.
Proper actions, fixedpoint algebras and naturality in nonabelian duality
 J. Funct. Anal
"... 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 C0(X). We consider a category in which the objects are C ∗dynamical systems (A, G, α) for which there is an equivariant homomorphism of (C0(X), γ) into the ..."
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Cited by 15 (8 self)
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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 C0(X). We consider a category in which the objects are C ∗dynamical systems (A, G, α) for which there is an equivariant homomorphism of (C0(X), γ) into the multiplier algebra M(A). Rieffel has shown that such systems are proper and saturated, and hence have a generalized fixedpoint algebra A α which is Morita equivalent to A×α,r G. We show that the assignment (A, α) ↦ → A α 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.
MANSFIELD’S IMPRIMITIVITY THEOREM FOR ARBITRARY CLOSED SUBGROUPS
, 2002
"... Let δ be a nondegenerate coaction of G on a C ∗algebra B, and let H be a closed subgroup of G. The dual action ˆ δ: H → Aut(B ×δ G) is proper and saturated in the sense of Rieffel, and the generalised fixedpoint algebra is the crossed product of B by the homogeneous space G/H. The resulting Morit ..."
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Cited by 13 (9 self)
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Let δ be a nondegenerate coaction of G on a C ∗algebra B, and let H be a closed subgroup of G. The dual action ˆ δ: H → Aut(B ×δ G) is proper and saturated in the sense of Rieffel, and the generalised fixedpoint algebra is the crossed product of B by the homogeneous space G/H. The resulting Morita equivalence is a version of Mansfield’s imprimitivity theorem which requires neither amenability nor normality of H.
Weight theory for C*algebraic quantum groups
, 1999
"... In this paper, we collect some technical results about weights on C∗algebras which are useful in de theory of locally compact quantum groups in the C∗algebra framework (see [17]). We discuss the extension of a lower semicontinuous weight to a normal weight following S. Baaj (see [1]), look into s ..."
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Cited by 9 (2 self)
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In this paper, we collect some technical results about weights on C∗algebras which are useful in de theory of locally compact quantum groups in the C∗algebra framework (see [17]). We discuss the extension of a lower semicontinuous weight to a normal weight following S. Baaj (see [1]), look into slice weights and their KSGNSconstructions and investigate the tensor product of weights together with a partial GNSconstruction for such a tensor product.
GENERALIZED FIXED POINT ALGEBRAS AND SQUAREINTEGRABLE GROUP ACTIONS
, 2000
"... Abstract. We analzye Rieffel’s construction of generalized fixed point algebras in the setting of group actions on Hilbert modules. Let G be a locally compact group acting on a C ∗algebra B. We construct a Hilbert module F over the reduced crossed product of G and B, using a pair (E, R), where E is ..."
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Cited by 8 (1 self)
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Abstract. We analzye Rieffel’s construction of generalized fixed point algebras in the setting of group actions on Hilbert modules. Let G be a locally compact group acting on a C ∗algebra B. We construct a Hilbert module F over the reduced crossed product of G and B, using a pair (E, R), where E is an equivariant Hilbert module over B and R is a dense subspace of E with certain properties. The generalized fixed point algebra is the C ∗algebra of compact operators on F. Any Hilbert module over the reduced crossed product arises by this construction for a pair (E, R) that is unique up to isomorphism. A necessary condition for the existence of R is that E be squareintegrable. The consideration of squareintegrable representations of Abelian groups on Hilbert space shows that this condition is not sufficient and that different choices for R may yield different generalized fixed point algebras. If B is proper in Kasparov’s sense, there is a unique R with the required properties. Thus the generalized fixed point algebra only depends on E. 1.
Proper Actions on Imprimitivity Bimodules and Decompositions of Morita Equivalences
, 2001
"... We consider a class of proper actions of locally compact groups on imprimitivity bimodules over C* ..."
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Cited by 8 (7 self)
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We consider a class of proper actions of locally compact groups on imprimitivity bimodules over C*
Coverings of directed graphs and crossed products of C ∗  algebras by coactions of homogeneous spaces
 Internat. J. Math
"... The CuntzKrieger algebra C ∗ (E) of a directed graph E is generated by a family of partial isometries satisfying relations which reflect the path structure of E. These graph algebras have a rich structure which is determined by the distribution of loops in the graph. Graph algebras have now arisen ..."
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Cited by 7 (5 self)
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The CuntzKrieger algebra C ∗ (E) of a directed graph E is generated by a family of partial isometries satisfying relations which reflect the path structure of E. These graph algebras have a rich structure which is determined by the distribution of loops in the graph. Graph algebras have now arisen in many different situations, and there is increasing interest in their interaction with other graphtheoretic ideas. Here we consider coverings of directed graphs: morphisms p: F → E of directed graphs which are local isomorphisms. We show that the graph algebra C ∗ (F) can be recovered from C ∗ (E) as a crossed product by a coaction of a homogeneous space associated to the fundamental group π1(E). The crossed products which arise this way are unusually tractable because we know so much about graph algebras, and in the second half of the paper we give some evidence that these crossed products of graph algebras will be good models for the general theory of crossed products by homogeneous spaces. Our results are motivated by work of Kumjian and Pask on regular coverings, which are the orbit maps p: F → E associated to free actions of a group G on a
CONTINUOUS SPECTRAL DECOMPOSITIONS OF ABELIAN GROUP ACTIONS ON C ∗ALGEBRAS
, 2007
"... Abstract. Let G be a locally compact Abelian group and ̂ G its Pontrjagin dual. Following Ruy Exel, we view Fell bundles over ̂ G as continuous spectral decompositions of Gactions on C ∗algebras. We classify such spectral decompositions using certain dense subspaces related to Marc Rieffel’s theor ..."
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Cited by 4 (4 self)
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Abstract. Let G be a locally compact Abelian group and ̂ G its Pontrjagin dual. Following Ruy Exel, we view Fell bundles over ̂ G as continuous spectral decompositions of Gactions on C ∗algebras. We classify such spectral decompositions using certain dense subspaces related to Marc Rieffel’s theory of squareintegrability. There is a unique continuous spectral decomposition if the group acts properly on the primitive ideal space of the C ∗algebra. But there are also examples of group actions without or with several inequivalent spectral decompositions. 1.
STRENGTH OF CONVERGENCE AND MULTIPLICITIES IN THE SPECTRUM OF A C ∗DYNAMICAL SYSTEM
, 2007
"... Abstract. We consider separable C ∗dynamical systems (A, G, α) for which the induced action of the group G on the primitive ideal space PrimA of the C ∗algebra A is free. We study how the representation theory of the associated crossedproduct C ∗algebra A ⋊α G depends on the representation theor ..."
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Cited by 4 (1 self)
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Abstract. We consider separable C ∗dynamical systems (A, G, α) for which the induced action of the group G on the primitive ideal space PrimA of the C ∗algebra A is free. We study how the representation theory of the associated crossedproduct C ∗algebra A ⋊α G depends on the representation theory of A and the properties of the action of G on PrimA and the spectrum Â. Our main tools involve computations of upper and lower bounds on multiplicity numbers associated to irreducible representations of A ⋊α G. We apply our techniques to give necessary and sufficient conditions, in terms of A and the action of G, for A ⋊α G to be (i) a continuoustrace C ∗algebra, (ii) a Fell C ∗algebra and (iii) a boundedtrace C ∗algebra. When G is amenable, we also give necessary and sufficient conditions for the crossedproduct C ∗algebra A ⋊α G to be (iv) a liminal C ∗algebra and (v) a Type I C ∗algebra. The results in (i), (iii)–(v) extend some earlier special cases in which A was assumed to have the corresponding property. 1.