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73
A class of C∗algebras generalizing both graph algebras and homeomorphism C∗algebras I, . . .
, 2003
"... We introduce a new class of C∗algebras, which is a generalization of both graph algebras and homeomorphism C ∗algebras. This class is very large and also very tractable. We prove the socalled gaugeinvariant uniqueness theorem and the CuntzKrieger uniqueness theorem, and compute the Kgroups of ..."
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Cited by 73 (7 self)
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We introduce a new class of C∗algebras, which is a generalization of both graph algebras and homeomorphism C ∗algebras. This class is very large and also very tractable. We prove the socalled gaugeinvariant uniqueness theorem and the CuntzKrieger uniqueness theorem, and compute the Kgroups of our algebras.
The ideal structure of C ∗  algebras of infinite graphs
 Illinois J. Math
"... Abstract. We classify the gaugeinvariant ideals in the C ∗algebras of infinite directed graphs, and describe the quotients as graph algebras. We then use these results to identify the gaugeinvariant primitive ideals in terms of the structural properties of the graph, and describe the Ktheory of ..."
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Cited by 63 (7 self)
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Abstract. We classify the gaugeinvariant ideals in the C ∗algebras of infinite directed graphs, and describe the quotients as graph algebras. We then use these results to identify the gaugeinvariant primitive ideals in terms of the structural properties of the graph, and describe the Ktheory of the C ∗algebras of arbitrary infinite graphs. 1.
Uniqueness theorems and ideal structure for Leavitt path algebras
, 2008
"... We prove Leavitt path algebra versions of the two uniqueness theorems of graph C ∗algebras. We use these uniqueness theorems to analyze the ideal structure of Leavitt path algebras and give necessary and sufficient conditions for their simplicity. We also use these results to give a proof of the ..."
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Cited by 58 (7 self)
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We prove Leavitt path algebra versions of the two uniqueness theorems of graph C ∗algebras. We use these uniqueness theorems to analyze the ideal structure of Leavitt path algebras and give necessary and sufficient conditions for their simplicity. We also use these results to give a proof of the fact that for any graph E the Leavitt path algebra LC(E) embeds as a dense ∗subalgebra of the graph C ∗algebra C ∗ (E). This embedding has consequences for graph C ∗algebras, and we discuss how we obtain new information concerning the construction of C ∗ (E).
Simplicity of C*algebras associated to higherrank graphs
 Bull. London Math. Soc
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Leavitt path algebras of arbitrary graphs
 HOUSTON J. MATH
, 2008
"... We extend the notion of the Leavitt path algebra of a graph to include all directed graphs. We show how various ringtheoretic properties of these more general structures relate to the corresponding properties of Leavitt path algebras of rowfinite graphs. Specifically, we identify those graphs fo ..."
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Cited by 42 (13 self)
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We extend the notion of the Leavitt path algebra of a graph to include all directed graphs. We show how various ringtheoretic properties of these more general structures relate to the corresponding properties of Leavitt path algebras of rowfinite graphs. Specifically, we identify those graphs for which the corresponding Leavitt path algebra is simple; purely infinite simple; exchange; and semiprime. In our final result, we show that all Leavitt path algebras have zero Jacobson radical.
IDEAL STRUCTURE OF C∗ALGEBRAS ASSOCIATED WITH C∗correspondences
, 2003
"... We study C∗algebras arising from C∗correspondences, which was introduced by the author. We prove the gaugeinvariant uniqueness theorem, and obtain conditions for our C∗algebras to be nuclear, exact, or satisfy the Universal Coefficient Theorem. We also obtain a 6term exact sequence of Kgroups ..."
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Cited by 29 (2 self)
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We study C∗algebras arising from C∗correspondences, which was introduced by the author. We prove the gaugeinvariant uniqueness theorem, and obtain conditions for our C∗algebras to be nuclear, exact, or satisfy the Universal Coefficient Theorem. We also obtain a 6term exact sequence of Kgroups involving the Kgroups of our C∗algebras.
ISOMORPHISM AND MORITA EQUIVALENCE OF GRAPH ALGEBRAS
, 2008
"... For any countable graph E, we investigate the relationship between the Leavitt path algebra LC(E) and the graph C∗algebra C∗(E). For graphs E and F, we examine ring homomorphisms, ring ∗homomorphisms, algebra homomorphisms, and algebra ∗homomorphisms between LC(E) and LC(F). We prove that in cer ..."
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Cited by 21 (7 self)
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For any countable graph E, we investigate the relationship between the Leavitt path algebra LC(E) and the graph C∗algebra C∗(E). For graphs E and F, we examine ring homomorphisms, ring ∗homomorphisms, algebra homomorphisms, and algebra ∗homomorphisms between LC(E) and LC(F). We prove that in certain situations isomorphisms between LC(E) and LC(F) yield ∗isomorphisms between the corresponding C∗algebras C ∗ (E) and C ∗ (F). Conversely, we show that∗isomorphisms between C ∗ (E) and C ∗ (F) produce isomorphisms between LC(E) and LC(F) in specific cases. The relationship between Leavitt path algebras and graph C ∗algebras is also explored in the context of Morita equivalence.
A unified approach to ExelLaca algebras and C∗algebras associated to graphs
 J. Operator Theory
"... Abstract. We define an ultragraph, which is a generalization of a directed graph, and describe how to associate a C∗algebra to it. We show that the class of ultragraph algebras contains the C∗algebras of graphs as well as the ExelLaca algebras. We also show that many of the techniques used for gr ..."
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Cited by 19 (3 self)
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Abstract. We define an ultragraph, which is a generalization of a directed graph, and describe how to associate a C∗algebra to it. We show that the class of ultragraph algebras contains the C∗algebras of graphs as well as the ExelLaca algebras. We also show that many of the techniques used for graph algebras can be applied to ultragraph algebras and that the ultragraph provides a useful tool for analyzing ExelLaca algebras. Our results include versions of the CuntzKrieger Uniqueness Theorem and the GaugeInvariant Uniqueness Theorem for ultragraph algebras.
Leavitt path algebras and direct limits
"... Abstract. An introduction to Leavitt path algebras of arbitrary directed graphs is presented, and direct limit techniques are developed, with which many results that had previously been proved for countable graphs can be extended to uncountable ones. Such results include characterizations of simplic ..."
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Cited by 18 (1 self)
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Abstract. An introduction to Leavitt path algebras of arbitrary directed graphs is presented, and direct limit techniques are developed, with which many results that had previously been proved for countable graphs can be extended to uncountable ones. Such results include characterizations of simplicity, characterizations of the exchange property, and cancellation conditions for the Ktheoretic monoid of equivalence classes of idempotent matrices.
A functorial approach to the C ∗ algebras of a graph
 Caterina Consani, Department of Mathematics, University of Toronto
"... ∗homomorphisms. The resulting C ∗algebras are identified as Toeplitz graph algebras. Graph algebras are proved to have inductive limit decompositions over any family of subgraphs with union equal to the whole graph. The construction is used to prove various structural properties of graph algebras. ..."
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Cited by 17 (5 self)
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∗homomorphisms. The resulting C ∗algebras are identified as Toeplitz graph algebras. Graph algebras are proved to have inductive limit decompositions over any family of subgraphs with union equal to the whole graph. The construction is used to prove various structural properties of graph algebras. Introduction. Since the paper of Cuntz and Krieger in 1976, much work has gone into elucidating the brief remarks made there regarding the case of infinite 0 − 1 matrices. While perhaps the most farreaching solution put forward has been a direct generalization to infinite 0 − 1 matrices ([9]), most of the papers on the subject generalize to a class of infinite directed graphs. (In fact, this is the direction indicated in [7].) In this paper