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15
Graphs, groupoids and CuntzKrieger algebras
, 1996
"... We associate to each locally finite directed graph G two locally compact groupoids G and G(?). The unit space of G is the space of onesided infinite paths in G, and G(?) is the reduction of G to the space of paths emanating from a distinguished vertex ?. We show that under certain conditions the ..."
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Cited by 43 (16 self)
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We associate to each locally finite directed graph G two locally compact groupoids G and G(?). The unit space of G is the space of onesided infinite paths in G, and G(?) is the reduction of G to the space of paths emanating from a distinguished vertex ?. We show that under certain conditions their C algebras are Morita equivalent; the groupoid C algebra C (G) is the CuntzKrieger algebra of an infinite f0; 1g matrix defined by G, and that the algebras C (G(?)) contain the C algebras used by Doplicher and Roberts in their duality theory for compact groups. We then analyse the ideal structure of these groupoid C algebras using the general theory of Renault, and calculate their Ktheory. 1 Introduction Over the past fifteen years many C algebras and classes of C algebras have been given groupoid models. Here we consider locally finite directed graphs, which may have infinitely many vertices, but only finitely many edges in and out of each vertex. We associate ...
C*Algebras of Directed Graphs and Group Actions
, 1997
"... Given a free action of a group G on a directed graph E we show that the crossed product of C*(E), the universal C*algebra of E, by the induced action is strongly Morita equivalent to C*(E/G). Since every connected graph E may be expressed as the quotient of a tree T by an action of a free group G ..."
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Cited by 30 (10 self)
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Given a free action of a group G on a directed graph E we show that the crossed product of C*(E), the universal C*algebra of E, by the induced action is strongly Morita equivalent to C*(E/G). Since every connected graph E may be expressed as the quotient of a tree T by an action of a free group G we may use our results to show that C*(E) is strongly Morita equivalent to the crossed product C0 (@T ) \Theta G, where @T is a certain 0dimensional space canonically associated to the tree.
A generalised GreenJulg theorem for proper groupoids and Banach algebras
 Preprintreihe SFB 478  Geometrische Strukturen in der Mathematik
, 2007
"... The GreenJulg theorem states that K G 0 (B) ∼ = K0(L 1 (G, B)) for every compact group G and every GC ∗algebra B. We formulate a generalisation of this result to proper groupoids and Banach algebras and deduce that the Bost assembly map is surjective for proper Banach algebras. Keywords: Green ..."
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Cited by 4 (4 self)
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The GreenJulg theorem states that K G 0 (B) ∼ = K0(L 1 (G, B)) for every compact group G and every GC ∗algebra B. We formulate a generalisation of this result to proper groupoids and Banach algebras and deduce that the Bost assembly map is surjective for proper Banach algebras. Keywords: GreenJulg theorem, locally compact groupoid, BaumConnes conjecture, Bost conjecture, Banach algebra;
Kduality for stratified pseudomanifolds
 MR2469513, Zbl 1159.19303
"... Abstract. This paper continues the project started in [13] where Poincaré duality in Ktheory was studied for singular manifolds with isolated conical singularities. Here, we extend the study and the results to general stratified pseudomanifolds. We review the axiomatic definition of a smooth strati ..."
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Cited by 3 (1 self)
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Abstract. This paper continues the project started in [13] where Poincaré duality in Ktheory was studied for singular manifolds with isolated conical singularities. Here, we extend the study and the results to general stratified pseudomanifolds. We review the axiomatic definition of a smooth stratification S of a topological space X and we define a groupoid T S X, called the Stangent space. This groupoid is made of different pieces encoding the tangent spaces of strata, and these pieces are glued into the smooth noncommutative groupoid T S X using the familiar procedure introduced by A. Connes for the tangent groupoid of a manifold. The main result is that C ∗ (T S X) is Poincaré dual to C(X), in other words, the Stangent space plays the role in Ktheory of a
Induction for Banach Algebras, Groupoids and KK ban
, 2008
"... Given two equivalent locally compact Hausdorff groupoids, the Bost conjecture with Banach algebra coefficients is true for one if and only if it is true for the other. This also holds for the Bost conjecture with C ∗coefficients. To show these results, the functoriality of Lafforgue’s KKtheory for ..."
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Cited by 1 (1 self)
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Given two equivalent locally compact Hausdorff groupoids, the Bost conjecture with Banach algebra coefficients is true for one if and only if it is true for the other. This also holds for the Bost conjecture with C ∗coefficients. To show these results, the functoriality of Lafforgue’s KKtheory for Banach algebras and groupoids with respect to generalised morphisms of groupoids is established. It is also shown that equivalent groupoids have Morita equivalent L 1algebras (with Banach algebra coefficients). Keywords: Locally compact groupoid, KK bantheory, Banach algebra, BaumConnes conjecture, Bost conjecture;
SELFSIMILAR GROUPS, ALGEBRAS AND SCHUR COMPLEMENTS
, 2006
"... Abstract. In the first part of the article we introduce C ∗algebras associated to selfsimilar groups and study their properties and relations to known algebras. The algebras are constructed as subalgebras of the CuntzPimsner algebra (and its homomorphic images) associated with the selfsimilarit ..."
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Abstract. In the first part of the article we introduce C ∗algebras associated to selfsimilar groups and study their properties and relations to known algebras. The algebras are constructed as subalgebras of the CuntzPimsner algebra (and its homomorphic images) associated with the selfsimilarity of the group. We study such properties as nuclearity, simplicity and Morita equivalence with algebras related to solenoids. The second part deals with the Schur complement transformations of elements of selfsimilar algebras. We study properties of such transformations and apply them to the spectral problem for Markov type elements in selfsimilar C ∗ −algebras. This is related to the spectral problem of the discrete Laplace operator on groups and graphs. Application of the Schur complement method in many situations reduces the spectral problem to study of invariant sets (very often of the type of a “strange attractor”) of a multidimensional rational transformation. A number of illustrating examples is provided. Finally we observe a relation between the Schur complement transformations and BartholdiKaimanovichVirag transformations of random walks on selfsimilar groups. 1.
Quasicrystals, C*algebras and Ktheory
, 2004
"... Quasicrystals are a phase between crystals and amorphous materials, exhibiting longrange order without periodicity. We review attempts to provide a theory of electronic transport in quasicrystals that may generalize Bloch theory. To this end, we introduce groupoid C*algebras, and use these to deve ..."
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Quasicrystals are a phase between crystals and amorphous materials, exhibiting longrange order without periodicity. We review attempts to provide a theory of electronic transport in quasicrystals that may generalize Bloch theory. To this end, we introduce groupoid C*algebras, and use these to develop Noncommutative Topology. This can be used to obtain a noncommutative C*algebra of observables in the aperiodic case that is a generalization of its periodic counterpart. The Ktheory of this C*algebra is used to obtain a labelling of the gaps in the spectrum of oneelectron Hamiltonians, and this labelling can be linked to the value of the integrated density of states on the gaps. Finally, we study a
C ∗pseudomultiplicative unitaries and Hopf
, 2009
"... We introduce C ∗pseudomultiplicative unitaries and concrete Hopf C ∗bimodules for the study of quantum groupoids in the setting of C ∗algebras. These unitaries and Hopf C ∗bimodules generalize multiplicative unitaries and Hopf C ∗algebras and are analogues of the pseudomultiplicative unitarie ..."
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We introduce C ∗pseudomultiplicative unitaries and concrete Hopf C ∗bimodules for the study of quantum groupoids in the setting of C ∗algebras. These unitaries and Hopf C ∗bimodules generalize multiplicative unitaries and Hopf C ∗algebras and are analogues of the pseudomultiplicative unitaries and Hopf–von Neumannbimodules studied by Enock, Lesieur and Vallin. To each C ∗pseudomultiplicative unitary, we associate two Fourier algebras with a duality pairing, a C ∗tensor category of representations, and in the regular case two reduced and two universal Hopf C ∗bimodules. The theory is illustrated by examples related to locally compact Hausdorff groupoids. In particular, we obtain a continuous Fourier algebra for a locally compact Hausdorff groupoid. 1