Abstract
We give necessary and sufficient conditions which a graph should satisfy in order for its associated C ∗-algebra to have a T 1 primitive ideal space. We give a description of which one-point sets in such a primitive ideal space are open, and use this to prove that anypurely infinite graph C ∗-algebra purely infinite graph C ∗-algebra purely infinite graph C ∗-algebra with a T 1 (in particular Hausdorff) primitive ideal space, is a c 0-direct sum of Kirchberg algebras. Moreover, we show that graph C ∗-algebras with a T 1 primitive ideal space canonically may be given the structure of a \(C(\tilde{\mathbb{N}})\)-algebra, and that isomorphisms of their \(\tilde{\mathbb{N}}\)-filtered K-theory (without coefficients) lift to \(E(\tilde{\mathbb{N}})\)-equivalences, as defined by Dadarlat and Meyer.
Mathematics Subject Classification (2010): 46L55, 46L35, 46M15, 19K35.
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Acknowledgements
The author would like to thank the referee for useful suggestions. The author would also like to thank his PhD-advisers Søren Eilers and Ryszard Nest for valuable discussions. This research was supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNFR92).
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Gabe, J. (2013). Graph C ∗-Algebras with a T 1 Primitive Ideal Space. In: Carlsen, T., Eilers, S., Restorff, G., Silvestrov, S. (eds) Operator Algebra and Dynamics. Springer Proceedings in Mathematics & Statistics, vol 58. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39459-1_7
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DOI: https://doi.org/10.1007/978-3-642-39459-1_7
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