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13
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 25 (11 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 ...
THE KTHEORY OF CUNTZKRIEGER ALGEBRAS FOR INFINITE MATRICES
, 1999
"... Abstract. We compute the Ktheory groups of the CuntzKrieger C ∗algebraOA associated to an infinite matrix A of zeros and ones. ..."
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Cited by 8 (0 self)
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Abstract. We compute the Ktheory groups of the CuntzKrieger C ∗algebraOA associated to an infinite matrix A of zeros and ones.
Computing Ext for graph algebras
 J. Operator Theory
"... Abstract. For a rowfinite graph G with no sinks and in which every loop has an exit, we construct an isomorphism between Ext(C ∗ (G)) and coker(A − I), where A is the vertex matrix of G. If c is the class in Ext(C ∗ (G)) associated to a graph obtained by attaching a sink to G, then this isomorphism ..."
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Cited by 5 (5 self)
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Abstract. For a rowfinite graph G with no sinks and in which every loop has an exit, we construct an isomorphism between Ext(C ∗ (G)) and coker(A − I), where A is the vertex matrix of G. If c is the class in Ext(C ∗ (G)) associated to a graph obtained by attaching a sink to G, then this isomorphism maps c to the class of a vector which describes how the sink was added. We conclude with an application in which we use this isomorphism to produce an example of a rowfinite transitive graph with no sinks whose associated C ∗algebra is not semiprojective. 1.
A family of 2graphs arising from twodimensional subshifts, Ergodic Theory Dynam
 Systems
"... Abstract. Higherrank graphs (or kgraphs) were introduced by Kumjian and Pask to provide combinatorial models for the higherrank CuntzKrieger C ∗algebras of Robertson and Steger. Here we consider a family of finite 2graphs whose path spaces are dynamical systems of algebraic origin, as studied ..."
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Cited by 1 (0 self)
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Abstract. Higherrank graphs (or kgraphs) were introduced by Kumjian and Pask to provide combinatorial models for the higherrank CuntzKrieger C ∗algebras of Robertson and Steger. Here we consider a family of finite 2graphs whose path spaces are dynamical systems of algebraic origin, as studied by Schmidt and others. We analyse the C ∗algebras of these 2graphs, find criteria under which they are simple and purely infinite, and compute their Ktheory. We find examples whose C ∗algebras satisfy the hypotheses of the classification theorem of Kirchberg and Phillips, but are not isomorphic to the C ∗algebras of ordinary directed graphs.
unknown title
, 1998
"... Abstract. Suppose a C ∗algebra A acts by adjointable operators on a Hilbert Amodule X. Pimsner constructed a C ∗algebraOX which includes, for particular choices of X, crossed products of A by Z, the Cuntz algebrasOn, and the CuntzKrieger algebrasOB. Here we analyse the representations of the cor ..."
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Abstract. Suppose a C ∗algebra A acts by adjointable operators on a Hilbert Amodule X. Pimsner constructed a C ∗algebraOX which includes, for particular choices of X, crossed products of A by Z, the Cuntz algebrasOn, and the CuntzKrieger algebrasOB. Here we analyse the representations of the corresponding Toeplitz algebra. One consequence is a uniqueness theorem for the ToeplitzCuntzKrieger algebras of directed graphs, which includes Cuntz’s uniqueness theorem forO∞. A Hilbert bimodule X over a C ∗algebra A is a right Hilbert Amodule with a left action of A by adjointable operators. The motivating example comes from an automorphism α of A: take XA = AA, and define the left action of A by a · b: = α(a)b. In [23], Pimsner constructed a C ∗algebra OX from a Hilbert bimodule X in such a way that the OX corresponding to an automorphism α is the crossed product A ×α Z. He also produced interesting examples of bimodules which do not arise from automorphisms or endomorphisms, including bimodules over finitedimensional commutative
unknown title
, 1998
"... Abstract. Suppose a C ∗algebra A acts by adjointable operators on a Hilbert Amodule X. Pimsner constructed a C ∗algebra OX which includes, for particular choices of X, crossed products of A by Z, the Cuntz algebras On, and the CuntzKrieger algebras OB. Here we analyse the representations of the ..."
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Abstract. Suppose a C ∗algebra A acts by adjointable operators on a Hilbert Amodule X. Pimsner constructed a C ∗algebra OX which includes, for particular choices of X, crossed products of A by Z, the Cuntz algebras On, and the CuntzKrieger algebras OB. Here we analyse the representations of the corresponding Toeplitz algebra. One consequence is a uniqueness theorem for the ToeplitzCuntzKrieger algebras of directed graphs, which includes Cuntz’s uniqueness theorem for O∞. A Hilbert bimodule X over a C ∗algebra A is a right Hilbert Amodule with a left action of A by adjointable operators. The motivating example comes from an automorphism α of A: take XA = AA, and define the left action of A by a · b: = α(a)b. In [23], Pimsner constructed a C ∗algebra OX from a Hilbert bimodule X in such a way that the OX corresponding to an automorphism α is the crossed product A ×α Z. He also produced interesting examples of bimodules which do not arise from automorphisms or endomorphisms, including bimodules over finitedimensional commutative
unknown title
, 1998
"... Abstract. Suppose a C ∗algebra A acts by adjointable operators on a Hilbert Amodule X. Pimsner constructed a C ∗algebraOX which includes, for particular choices of X, crossed products of A by Z, the Cuntz algebrasOn, and the CuntzKrieger algebrasOB. Here we analyse the representations of the cor ..."
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Abstract. Suppose a C ∗algebra A acts by adjointable operators on a Hilbert Amodule X. Pimsner constructed a C ∗algebraOX which includes, for particular choices of X, crossed products of A by Z, the Cuntz algebrasOn, and the CuntzKrieger algebrasOB. Here we analyse the representations of the corresponding Toeplitz algebra. One consequence is a uniqueness theorem for a family of ToeplitzCuntzKrieger algebras for directed graphs, which includes Cuntz’s uniqueness theorem forO∞. A Hilbert bimodule X over a C ∗algebra A is a right Hilbert Amodule with a left action of A by adjointable operators. The motivating example comes from an automorphism α of A: take XA = AA, and define the left action of A by a · b: = α(a)b. In [23], Pimsner constructed a C ∗algebra OX from a Hilbert bimodule X in such a way that the OX corresponding to an automorphism α is the crossed product A ×α Z. He also produced interesting examples of bimodules which do not arise from automorphisms or endomorphisms, including bimodules over finitedimensional commutative
TOPOLOGICAL QUIVERS
, 2005
"... Abstract. Topological quivers are generalizations of directed graphs in which the sets of vertices and edges are locally compact Hausdorff spaces. Associated to such a topological quiver Q is a C ∗correspondence, and from this correspondence one may construct a CuntzPimsner algebra C ∗ (Q). In thi ..."
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Abstract. Topological quivers are generalizations of directed graphs in which the sets of vertices and edges are locally compact Hausdorff spaces. Associated to such a topological quiver Q is a C ∗correspondence, and from this correspondence one may construct a CuntzPimsner algebra C ∗ (Q). In this paper we develop the general theory of topological quiver C ∗algebras and show how certain C ∗algebras found in the literature may be viewed from this general perspective. In particular, we show that C ∗algebras of topological quivers generalize the wellstudied class of graph C ∗algebras and in analogy with that theory much of the operator algebra structure of C ∗ (Q) can be determined fromQ. We also show that many fundamental results from the theory of graph C ∗algebras have natural analogues in the context of topological quivers (often with more involved proofs). These include the GaugeInvariant Uniqueness theorem, the CuntzKrieger Uniqueness theorem, descriptions of the ideal structure, and conditions for simplicity. 1.
Examining Committee:
, 2002
"... This thesis represents the efforts of many individuals besides myself whom I would like to acknowledge and also thank. In completing this work I was helped by numerous people in a myriad of ways. It is a testament to the amount of assistance and support that I received that I am unable to list all o ..."
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This thesis represents the efforts of many individuals besides myself whom I would like to acknowledge and also thank. In completing this work I was helped by numerous people in a myriad of ways. It is a testament to the amount of assistance and support that I received that I am unable to list all of the contributors in a few pages. I therefore apologize in advance for the incompleteness of the following list. First and foremost I would like to thank my advisor, the indefatigable Dana Williams, for his patience, support, and guidance. His knowledge and insight have been excellent resources for me, and his friendliness and freeflowing advice have been great assets to me during my graduate career. He has always provided the right balance of assistance, pressure, and encouragement — helping me when I needed it and leaving me on my own when I needed to figure things out for myself. I would also like to thank Iain Raeburn for his many valuable suggestions, countless insights, and constant support. His expertise has been a great source of inspiration for me. In addition, he has assisted me more times than I can count, and I am very grateful to him for it. I would also like to thank him for giving me the opportunity to