Results 1  10
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20
Higher rank graph C*algebras
, 2000
"... Building on recent work of Robertson and Steger, we associate a C ∗ –algebra to a combinatorial object which may be thought of as a higher rank graph. This C ∗ –algebra is shown to be isomorphic to that of the associated path groupoid. Various results in this paper give sufficient conditions on the ..."
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Cited by 58 (11 self)
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Building on recent work of Robertson and Steger, we associate a C ∗ –algebra to a combinatorial object which may be thought of as a higher rank graph. This C ∗ –algebra is shown to be isomorphic to that of the associated path groupoid. Various results in this paper give sufficient conditions on the higher rank graph for the associated C ∗ –algebra to be: simple, purely infinite and AF. Results concerning the structure of crossed products by certain natural actions of discrete groups are obtained; a technique for constructing rank 2 graphs from “commuting” rank 1 graphs is given.
Analytic algebras of higher rank graphs
 Math. Proc. Royal Irish Acad
"... Abstract. We begin the study of a new class of operator algebras that arise from higher rank graphs. Every higher rank graph generates a Fock space Hilbert space and creation operators which are partial isometries acting on the space. We call the weak operator topology closed algebra generated by th ..."
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Cited by 24 (9 self)
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Abstract. We begin the study of a new class of operator algebras that arise from higher rank graphs. Every higher rank graph generates a Fock space Hilbert space and creation operators which are partial isometries acting on the space. We call the weak operator topology closed algebra generated by these operators a higher rank semigroupoid algebra. A number of examples are discussed in detail, including the single vertex case and higher rank cycle graphs. In particular the cycle graph algebras are identified as matricial multivariable function algebras. We obtain reflexivity for a wide class of graphs and characterize semisimplicity in terms of the underlying graph. In [22] Kumjian and Pask introduced kgraphs as an abstraction of the combinatorial structure underlying the higher rank graph C ∗algebras of Robertson and Steger [31, 32]. A kgraph generalizes the set of finite paths of a countable directed graph when viewed as a partly defined multiplicative semigroup with vertices considered as degenerate paths. The C ∗algebras associated with kgraphs include kfold tensor products of graph C ∗algebras, and much more [2, 21, 26, 27, 30]. On the other hand, as a generalization of the nonselfadjoint free semigroup
Higher rank graph C∗algebras
, 2000
"... Building on recent work of Robertson and Steger, we associate a C*algebra to a combinatorial object which maybe thought of as a higher rank graph. This C*algebra is shown to be isomorphic to that of the associated path groupoid. Various results in this paper give sufficient conditions on the highe ..."
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Cited by 14 (2 self)
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Building on recent work of Robertson and Steger, we associate a C*algebra to a combinatorial object which maybe thought of as a higher rank graph. This C*algebra is shown to be isomorphic to that of the associated path groupoid. Various results in this paper give sufficient conditions on the higher rank graph for the associated C*algebra to be: simple, purely infinite and AF. Results concerning the structure of crossed products by certain natural actions of discrete groups are obtained; a technique for constructing rank 2 graphs from "commuting" rank 1 graphs is given.
Asymptotic Ktheory for groups acting on Ã2 buildings
 Canad. J. Math
"... Abstract. Let Γ be a torsion free lattice in G = PGL(3, F) where F is a nonarchimedean local field. Then Γ acts freely on the affine BruhatTits building B of G and there is an induced action on the boundary Ω of B. The crossed product C ∗algebra A(Γ) = C(Ω) ⋊Γ depends only on Γ and is classified b ..."
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Cited by 13 (2 self)
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Abstract. Let Γ be a torsion free lattice in G = PGL(3, F) where F is a nonarchimedean local field. Then Γ acts freely on the affine BruhatTits building B of G and there is an induced action on the boundary Ω of B. The crossed product C ∗algebra A(Γ) = C(Ω) ⋊Γ depends only on Γ and is classified by its Ktheory. This article shows how to compute the Ktheory of A(Γ) and of the larger class of rank two CuntzKrieger algebras. 1.
Classifying higher rank analytic Toeplitz algebras
"... Abstract. To a higher rank directed graph (Λ, d), in the sense of Kumjian and Pask [16], one can associate natural noncommutative analytic Toeplitz algebras, both weakly closed and norm closed. We introduce methods for the classification of these algebras in the case of single vertex graphs. 1. ..."
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Cited by 12 (3 self)
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Abstract. To a higher rank directed graph (Λ, d), in the sense of Kumjian and Pask [16], one can associate natural noncommutative analytic Toeplitz algebras, both weakly closed and norm closed. We introduce methods for the classification of these algebras in the case of single vertex graphs. 1.
Higherrank graph C∗algebras: an inverse semigroup and groupoid approach
, 2006
"... We provide inverse semigroup and groupoid models for the Toeplitz and CuntzKrieger algebras of finitely aligned higherrank graphs. Using these models, we prove a uniqueness theorem for the CuntzKrieger algebra. ..."
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Cited by 11 (4 self)
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We provide inverse semigroup and groupoid models for the Toeplitz and CuntzKrieger algebras of finitely aligned higherrank graphs. Using these models, we prove a uniqueness theorem for the CuntzKrieger algebra.
On higher rank graph C ∗ algebras
, 2002
"... Abstract. Given a rowfinite kgraph Λ with no sources we investigate the Ktheory of the higher rank graph C ∗algebra, C ∗ (Λ). When k = 2 we are able to give explicit formulae to calculate the Kgroups of C ∗ (Λ). The Kgroups of C ∗ (Λ) for k> 2 can be calculated under certain circumstances. ..."
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Cited by 8 (1 self)
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Abstract. Given a rowfinite kgraph Λ with no sources we investigate the Ktheory of the higher rank graph C ∗algebra, C ∗ (Λ). When k = 2 we are able to give explicit formulae to calculate the Kgroups of C ∗ (Λ). The Kgroups of C ∗ (Λ) for k> 2 can be calculated under certain circumstances. We state that for all k, the torsionfree rank of K0(C ∗ (Λ)) and K1(C ∗ (Λ)) are equal when C ∗ (Λ) is unital, and we determine the position of the class of the unit of C ∗ (Λ) in K0(C ∗ (Λ)). 1.
Boundary actions for affine buildings and higher rank CuntzKrieger algebras. C∗algebras
 Proceedings of the SFB Workshop on C∗algebras (Münster, March 8–12, 1999), 182–202, SpringerVerlag, 2000, MR 2001j:46082
"... Abstract. Let Γ be a group of type rotating automorphisms of an affine building B of type Ã2. If Γ acts freely on the vertices of B with finitely many orbits, and if Ω is the (maximal) boundary of B, then C(Ω) ⋊ Γ is a p.i.s.u.n. C ∗algebra. This algebra has a structure theory analogous to that of ..."
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Cited by 7 (2 self)
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Abstract. Let Γ be a group of type rotating automorphisms of an affine building B of type Ã2. If Γ acts freely on the vertices of B with finitely many orbits, and if Ω is the (maximal) boundary of B, then C(Ω) ⋊ Γ is a p.i.s.u.n. C ∗algebra. This algebra has a structure theory analogous to that of a simple CuntzKrieger algebra and is the motivation for a theory of higher rank CuntzKrieger algebras, which has been developed by T. Steger and G. Robertson. The Ktheory of these algebras can be computed explicitly in the rank two case. For the rank two examples of the form C(Ω) ⋊ Γ which arise from boundary actions on Ã2 buildings, the two Kgroups coincide.
Higherrank graphs and their C ∗ algebras
 MR1961175 (2004f:46068), Zbl pre01925883
"... Abstract. We consider the higherrank graphs introduced by Kumjian and Pask as models for higherrank CuntzKrieger algebras. We describe a variant of the CuntzKrieger relations which applies to graphs with sources, and describe a local convexity condition which characterises the higherrank graphs ..."
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Cited by 5 (2 self)
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Abstract. We consider the higherrank graphs introduced by Kumjian and Pask as models for higherrank CuntzKrieger algebras. We describe a variant of the CuntzKrieger relations which applies to graphs with sources, and describe a local convexity condition which characterises the higherrank graphs that admit a nontrivial CuntzKrieger family. We then prove versions of the uniqueness theorems and classifications of ideals for the C ∗algebras generated by CuntzKrieger families. 1.
Fundamental groupoids of kgraphs
, 2003
"... Abstract. kgraphs are higherrank analogues of directed graphs which were first developed to provide combinatorial models for operator algebras of CuntzKrieger type. Here we develop a theory of the fundamental groupoid of a kgraph, and relate it to the fundamental groupoid of an associated graph ..."
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Cited by 4 (2 self)
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Abstract. kgraphs are higherrank analogues of directed graphs which were first developed to provide combinatorial models for operator algebras of CuntzKrieger type. Here we develop a theory of the fundamental groupoid of a kgraph, and relate it to the fundamental groupoid of an associated graph called the 1skeleton. We also explore the failure, in general, of kgraphs to faithfully embed into their fundamental groupoids. 1.