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TOPOLOGICAL HIGHERRANK GRAPHS AND THE C ∗ALGEBRAS OF TOPOLOGICAL 1GRAPHS
, 2005
"... Abstract. We introduce the notion of a topological higherrank graph, a unified generalization of the higherrank graph and the topological graph. Using groupoid techniques, we define the Toeplitz and CuntzKrieger algebras of topological higherrank graphs, and show that the C ∗algebras defined ar ..."
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Abstract. We introduce the notion of a topological higherrank graph, a unified generalization of the higherrank graph and the topological graph. Using groupoid techniques, we define the Toeplitz and CuntzKrieger algebras of topological higherrank graphs, and show that the C ∗algebras defined are coherent with the existing theory. 1.
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 conside ..."
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Cited by 2 (0 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
RANKTWO GRAPHS WHOSE C ∗ALGEBRAS ARE DIRECT LIMITS OF CIRCLE ALGEBRAS
, 2005
"... Abstract. We describe a class of rank2 graphs whose C ∗algebras are AT algebras. For a subclass which we call rank2 Bratteli diagrams, we compute the Ktheory of the C ∗algebra. We identify rank2 Bratteli diagrams whose C ∗algebras are simple and have realrank zero, and characterise the Kinv ..."
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Cited by 2 (1 self)
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Abstract. We describe a class of rank2 graphs whose C ∗algebras are AT algebras. For a subclass which we call rank2 Bratteli diagrams, we compute the Ktheory of the C ∗algebra. We identify rank2 Bratteli diagrams whose C ∗algebras are simple and have realrank zero, and characterise the Kinvariants achieved by such algebras. We give examples of rank2 Bratteli diagrams whose C ∗algebras contain as full corners the irrational rotation algebras and the BunceDeddens algebras. 1.
C ∗ALGEBRAS ASSOCIATED TO COVERINGS OF kGRAPHS
, 2006
"... Abstract. A covering of kgraphs (in the sense of PaskQuiggRaeburn) induces an embedding of universal C ∗algebras. We show how to build a (k + 1)graph whose universal algebra encodes this embedding. More generally we show how to realise a direct limit of kgraph algebras under embeddings induced ..."
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Cited by 1 (1 self)
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Abstract. A covering of kgraphs (in the sense of PaskQuiggRaeburn) induces an embedding of universal C ∗algebras. We show how to build a (k + 1)graph whose universal algebra encodes this embedding. More generally we show how to realise a direct limit of kgraph algebras under embeddings induced from coverings as the universal algebra of a (k + 1)graph. Our main focus is on computing the Ktheory of the (k+1)graph algebra from that of the component kgraph algebras. Examples of our construction include a realisation of the Kirchberg algebra Pn whose Ktheory is opposite to that of On, and a class of ATalgebras that can naturally be regarded as higherrank BunceDeddens algebras. 1.
GRAPHS OF C ∗CORRESPONDENCES AND FELL BUNDLES
"... Abstract. We define the notion of a Λsystem of C ∗correspondences associated to a higherrank graph Λ. Roughly speaking, such a system assigns to each vertex of Λ a C ∗algebra, and to each path in Λ a C ∗correspondence in a way which carries compositions of paths to balanced tensor products of C ..."
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Abstract. We define the notion of a Λsystem of C ∗correspondences associated to a higherrank graph Λ. Roughly speaking, such a system assigns to each vertex of Λ a C ∗algebra, and to each path in Λ a C ∗correspondence in a way which carries compositions of paths to balanced tensor products of C ∗correspondences. Under some simplifying assumptions, we use Fowler’s technology of CuntzPimsner algebras for product systems of C ∗correspondences to associate a C ∗algebra to each Λsystem. We then construct a Fell bundle over the path groupoid GΛ and show that the C ∗algebra of the Λsystem coincides with the reduced crosssectional algebra of the Fell bundle. We conclude by discussing several examples of our construction arising in the literature. 1.
C∗Algebras Associated to Coverings of kGraphs
 DOCUMENTA MATH.
, 2007
"... A covering of kgraphs (in the sense of PaskQuiggRaeburn) induces an embedding of universal C∗algebras. We show how to build a (k + 1)graph whose universal algebra encodes this embedding. More generally we show how to realise a direct limit of kgraph algebras under embeddings induced from cover ..."
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A covering of kgraphs (in the sense of PaskQuiggRaeburn) induces an embedding of universal C∗algebras. We show how to build a (k + 1)graph whose universal algebra encodes this embedding. More generally we show how to realise a direct limit of kgraph algebras under embeddings induced from coverings as the universal algebra of a (k + 1)graph. Our main focus is on computing the Ktheory of the (k+1)graph algebra from that of the component kgraph algebras. Examples of our construction include a realisation of the Kirchberg algebra Pn whose Ktheory is opposite to that of On, and a class of ATalgebras that can naturally be regarded as higherrank BunceDeddens algebras.
HOMOLOGY FOR HIGHERRANK GRAPHS AND TWISTED C ∗ALGEBRAS
"... Abstract. We introduce a homology theory for kgraphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a kgraph coincides with the homology of its topological realisation as described by Kaliszewski et al. We exhibit combinator ..."
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Abstract. We introduce a homology theory for kgraphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a kgraph coincides with the homology of its topological realisation as described by Kaliszewski et al. We exhibit combinatorial versions of a number of standard topological constructions, and show that they are compatible, from a homological point of view, with their topological counterparts. We show how to twist the C ∗algebra of a kgraph by a Tvalued 2cocycle and demonstrate that examples include all noncommutative tori. In the appendices, we construct a cubical set ˜ Q(Λ) from a kgraph Λ and demonstrate that the homology and topological realisation of Λ coincide with those of ˜ Q(Λ) as defined by Grandis. 1.