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On the Primitive Ideal spaces of the $C^*$-algebras of graphs

Published online by Cambridge University Press:  21 October 2005

TERESA BATES
Affiliation:
School of Mathematics, The University of New South Wales, UNSW Sydney 2052, Australia. e-mail: [email protected]

Abstract

We characterise the topological spaces which arise as the primitive ideal spaces of the Cuntz–Krieger algebras of graphs satisfying condition (K): directed graphs in which every vertex lying on a loop lies on at least two loops. We deduce that the spaces which arise as ${\rm Prim}\;C^*(E)$ are precisely the spaces which arise as the primitive ideal spaces of AF-algebras. Finally, we construct a graph $\wt{E}$ from E such that $C^*(\wt{E})$ is an AF-algebra and ${\rm Prim}\;C^*(E)$ and ${\rm Prim}\;C^*(\wt{E})$ are homeomorphic.

Type
Research Article
Copyright
2005 Cambridge Philosophical Society

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