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

@article{Katsura2006ACO, title={A class of \$\{C^*\}\$-algebras generalizing both graph algebras and homeomorphism \$\{C^*\}\$-algebras III, ideal structures}, author={Takeshi Katsura}, journal={Ergodic Theory and Dynamical Systems}, year={2006}, volume={26}, pages={1805 - 1854} }

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 action, and give a condition on topological graphs so that all ideals are invariant under the gauge action. We get conditions for our $C^*$-algebras to be simple, prime or primitive. We completely determine the prime ideals, and show that most of them are primitive. Finally, we construct a discrete… Expand

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#### References

SHOWING 1-10 OF 23 REFERENCES

The $C^*$-Algebras of Arbitrary Graphs

- Mathematics
- 2000

To an arbitrary directed graph we associate a row-finite directed graph whose C*-algebra contains the C*-algebra of the original graph as a full corner. This allows us to generalize results for… Expand

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

- Mathematics
- 2004

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… Expand

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

- Mathematics
- 2002

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… Expand

The ideal structure of the $C\sp *$-algebras of infinite graphs

- Mathematics
- 2001

We classify the gauge-invariant ideals in the C*-algebras of infinite directed graphs, and describe the quotients as graph algebras. We then use these results to identify the gauge-invariant… Expand

Ideal structure of C∗-algebras associated with C∗-correspondences

- Mathematics
- 2003

We study the ideal structure of C*-algebras arising from C*-correspondences. We prove that gauge-invariant ideals of our C*-algebras are parameterized by certain pairs of ideals of original… Expand

The Topology on the Primitive Ideal Space of Transformation Group C # - Algebras and C.C.R. Transformation Group C # -Algebras

- Mathematics
- 1981

If (G, 8) is a second countable transformation group and the stability groups are amenable then C*(G, 8) is C.C.R. if and only if the orbits are closed and the stability groups are C.C.R. In… Expand

Topological Quivers

- Mathematics
- 2003

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… Expand

THE C -ALGEBRAS OF ROW-FINITE GRAPHS

- Mathematics
- 2000

NSKI Abstract. We prove versions of the fundamental theorems about Cuntz-Krieger algebras for the C -algebras of row-finite graphs: directed graphs in which each vertex emits at most finitely many… Expand

Morita Equivalence and Continuous-Trace $C^*$-Algebras

- Mathematics
- 1998

The algebra of compact operators Hilbert $C^*$-modules Morita equivalence Sheaves, cohomology, and bundles Continuous-trace $C^*$-algebras Applications Epilogue: The Brauer group and group actions… Expand

Graphs, Groupoids, and Cuntz–Krieger Algebras

- Mathematics
- 1997

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… Expand