Ultragraph C -algebras via topological quivers

@article{Katsura2006UltragraphC,
  title={Ultragraph C -algebras via topological quivers},
  author={Takeshi Katsura and Paul S. Muhly and Aidan Sims and Mark Tomforde},
  journal={Studia Mathematica},
  year={2006},
  volume={187},
  pages={137-155}
}
Given an ultragraph in the sense of Tomforde, we construct a topological quiver in the sense of Muhly and Tomforde in such a way that the universal C -algebras associated to the two objects coincide. We apply results of Muhly and Tomforde for topological quiver algebras and of Katsura for topological graph C -algebras to study the K-theory and gauge-invariant ideal structure of ultragraph C -algebras. 
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21 Citations
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Background Citations
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Methods Citations
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21 Citations

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References

SHOWING 1-10 OF 25 REFERENCES

Simplicity of ultragraph algebras

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…

A CLASS OF C*-ALGEBRAS GENERALIZING BOTH GRAPH ALGEBRAS AND HOMEOMORPHISM C*-ALGEBRAS II, EXAMPLES

We show that the method to construct C*-algebras from topological graphs, introduced in our previous paper, generalizes many known constructions. We give many ways to make new topological graphs from…

A unified approach to Exel-Laca algebras and C*-algebras associated to graphs

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…

Topological Quivers

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…

A class of C*-algebras generalizing both graph algebras and homeomorphism C*-algebras I, fundamental results

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 so-called…

GRAPH INVERSE SEMIGROUPS, GROUPOIDS AND THEIR C -ALGEBRAS

We develop a theory of graph C -algebras using path groupoids and inverse semigroups. Row finiteness is not assumed so that the theory applies to graphs for which there are vertices emitting a…

Higher-Rank Graph C *-Algebras: An Inverse Semigroup and Groupoid Approach

AbstractWe provide inverse semigroup and groupoid models for the Toeplitz and Cuntz-Krieger algebras of finitely aligned higher-rank graphs. Using these models, we prove a uniqueness theorem for the…

A class ofC*-algebras and topological Markov chains

In this paper we present a class of C*-algebras and point out its close relationship to topological Markov chains, whose theory is part of symbolic dynamics. The C*-algebra construction starts from a…

Graphs, Groupoids, and Cuntz–Krieger Algebras

We associate to each locally finite directed graphGtwo locally compact groupoidsGandG(★). The unit space ofGis the space of one–sided infinite paths inG, andG(★) is the reduction ofGto the space of…

A class of ${C^*}$-algebras generalizing both graph algebras and homeomorphism ${C^*}$-algebras III, ideal structures

We investigate the ideal structures of the $C^*$-algebras arising from topological graphs. We give a complete description of ideals of such $C^*$-algebras that are invariant under the so-called gauge…