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41
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 126 (32 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.
Affine buildings, tiling systems and higher rank CuntzKrieger algebras
, 1999
"... Abstract. To an rdimensional subshift of finite type satisfying certain special properties we associate a C∗algebra A. This algebra is a higher rank version of a CuntzKrieger algebra. In particular, it is simple, purely infinite and nuclear. We study an example: if Γ is a group acting freely on t ..."
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Cited by 97 (13 self)
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Abstract. To an rdimensional subshift of finite type satisfying certain special properties we associate a C∗algebra A. This algebra is a higher rank version of a CuntzKrieger algebra. In particular, it is simple, purely infinite and nuclear. We study an example: if Γ is a group acting freely on the vertices of an Ã2 building, with finitely many orbits, and if Ω is the boundary of that building, then C(Ω)o Γ is the algebra associated to a certain two dimensional subshift.
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 62 (14 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 36 (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
On the Ktheory of higher rank graph C*algebras
, 2008
"... 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 and we cons ..."
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Cited by 32 (1 self)
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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 and we consider the case k =3. Weprovethat for arbitrary k, the torsionfree rank of K0(C ∗ (Λ)) and K1(C ∗ (Λ)) are equal when C ∗ (Λ) is unital, and for k = 2 we determine the position of
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 28 (6 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 18 (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.
Fundamental groupoids of kgraphs
, 2004
"... 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 14 (12 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.
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 12 (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 10 (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.