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51
A class of C∗algebras generalizing both graph algebras and homeomorphism C∗algebras I, Fundamental results
, 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 C ∗ algebras of arbitrary graphs
 Rocky Mountain J. Math
"... Abstract. To an arbitrary directed graph we associate a rowfinite directed graph whose C ∗algebra contains the C ∗algebra of the original graph as a full corner. This allows us to generalize results for C ∗algebras of rowfinite graphs to C ∗algebras of arbitrary graphs: the uniqueness theorem, ..."
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Cited by 72 (30 self)
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Abstract. To an arbitrary directed graph we associate a rowfinite directed graph whose C ∗algebra contains the C ∗algebra of the original graph as a full corner. This allows us to generalize results for C ∗algebras of rowfinite graphs to C ∗algebras of arbitrary graphs: the uniqueness theorem, simplicity criteria, descriptions of the ideals and primitive ideal space, and conditions under which a graph algebra is AF and purely infinite. Our proofs require only standard CuntzKrieger techniques and do not rely on powerful constructs such as groupoids, ExelLaca algebras, or CuntzPimsner algebras. 1.
The ideal structure of C∗algebras of infinite graphs
 ILLINOIS J. MATH
, 2001
"... 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 ∗al ..."
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Cited by 60 (4 self)
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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.
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 (15 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.
Inverse semigroups and combinatorial C*algebras
, 2008
"... We describe a special class of representations of an inverse semigroup S on Hilbert’s space which we term tight. These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the tight spectrum, which is in turn shown to be precisely the closure of the sp ..."
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Cited by 30 (7 self)
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We describe a special class of representations of an inverse semigroup S on Hilbert’s space which we term tight. These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the tight spectrum, which is in turn shown to be precisely the closure of the space of ultrafilters, once filters are identified with semicharacters in a natural way. These representations are moreover shown to correspond to representations of the C*algebra of the groupoid of germs for the action of S on its tight spectrum. We then treat the case of certain inverse semigroups constructed from semigroupoids, generalizing and inspired by inverse semigroups constructed from ordinary and higher rank graphs. The tight representations of this inverse semigroup are in onetoone correspondence with representations of the semigroupoid, and consequently the semigroupoid algebra is given a groupoid model. The groupoid which arises from this construction is shown to be the same as the boundary path groupoid of Farthing, Muhly and Yeend, at least in the singly aligned, sourceless case.
THE PRIMITIVE IDEAL SPACE OF THE C∗ALGEBRAS OF INFINITE GRAPHS
, 2002
"... For any countable directed graph E we describe the primitive ideal space of the corresponding generalized CuntzKrieger algebra C∗(E). ..."
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Cited by 24 (1 self)
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For any countable directed graph E we describe the primitive ideal space of the corresponding generalized CuntzKrieger algebra C∗(E).
Simplicity of ultragraph algebras
 Indiana Univ. Math. J
, 2001
"... Abstract. In this paper we analyze the structure of C ∗algebras associated to ultragraphs, which are generalizations of directed graphs. We characterize the simple ultragraph algebras as well as deduce necessary and sufficient conditions for an ultragraph algebra to be purely infinite and to be AF. ..."
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Cited by 14 (8 self)
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Abstract. In this paper we analyze the structure of C ∗algebras associated to ultragraphs, which are generalizations of directed graphs. We characterize the simple ultragraph algebras as well as deduce necessary and sufficient conditions for an ultragraph algebra to be purely infinite and to be AF. Using these techniques we also produce an example of an ultragraph algebra that is neither a graph algebra nor an ExelLaca algebra. We conclude by proving that the C ∗algebras of ultragraphs with no sinks are CuntzPimsner algebras. 1.
A GROUPOID APPROACH TO DISCRETE INVERSE SEMIGROUP ALGEBRAS
, 2009
"... Let K be a commutative ring with unit and S an inverse semigroup. We show that the semigroup algebra KS can be described as a convolution algebra of functions on the universal étale groupoid associated to S by Paterson. This result is a simultaneous generalization of the author’s earlier work on fin ..."
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Cited by 14 (4 self)
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Let K be a commutative ring with unit and S an inverse semigroup. We show that the semigroup algebra KS can be described as a convolution algebra of functions on the universal étale groupoid associated to S by Paterson. This result is a simultaneous generalization of the author’s earlier work on finite inverse semigroups and Paterson’s theorem for the universal C∗algebra. It provides a convenient topological framework for understanding the structure of KS, including the center and when it has a unit. In this theory, the role of Gelfand duality is replaced by Stone duality. Using this approach we are able to construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups, generalizing the wellstudied case of an inverse semigroup with finitely many idempotents. More generally, we describe the irreducible representations of an inverse semigroup S that can be induced from associated groups as precisely those satisfying a certain “finiteness condition”. This “finiteness condition ” is satisfied, for instance, by all representations of an inverse semigroup whose image contains a primitive idempotent.