Abstract
This survey reports on current progress of programs to classify graph C ∗-algebras and Leavitt path algebras up to Morita equivalence using K-theory. Beginning with an overview and some history, we trace the development of the classification of simple and nonsimple graph C ∗-algebras and state theorems summarizing the current status of these efforts. We then discuss the much more nascent efforts to classify Leavitt path algebras, and we describe the current status of these efforts as well as outline current impediments that must be solved for this classification program to progress. In particular, we give two specific open problems that must be addressed in order to identify the correct K-theoretic invariant for classification of simple Leavitt path algebras, and we discuss the significance of various possible outcomes to these open problems.
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Acknowledgements
The author thanks the organizers of the 2015 Abel Symposium, Christian Skau (Norwegian University of Science and Technology), Toke M. Carlsen (University of the Faroe Islands), Nadia Larsen (University of Oslo) and Sergey Neshveyev (University of Oslo) for their hospitality and the opportunity to attend. This work, including the author’s travel to the Abel Symposium, was supported by a grant from the Simons Foundation (#210035 to Mark Tomforde).
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Tomforde, M. (2016). Classification of Graph Algebras: A Selective Survey. In: Carlsen, T.M., Larsen, N.S., Neshveyev, S., Skau, C. (eds) Operator Algebras and Applications. Abel Symposia, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-39286-8_14
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DOI: https://doi.org/10.1007/978-3-319-39286-8_14
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