Generalised morphisms of $k$-graphs: $k$-morphs
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- by Alex Kumjian, David Pask and Aidan Sims
- Trans. Amer. Math. Soc. 363 (2011), 2599-2626
- DOI: https://doi.org/10.1090/S0002-9947-2010-05152-9
- Published electronically: December 20, 2010
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Abstract:
In a number of recent papers, $(k+l)$-graphs have been constructed from $k$-graphs by inserting new edges in the last $l$ dimensions. These constructions have been motivated by $C^*$-algebraic considerations, so they have not been treated systematically at the level of higher-rank graphs themselves. Here we introduce $k$-morphs, which provide a systematic unifying framework for these various constructions. We think of $k$-morphs as the analogue, at the level of $k$-graphs, of $C^*$-correspondences between $C^*$-algebras. To make this analogy explicit, we introduce a category whose objects are $k$-graphs and whose morphisms are isomorphism classes of $k$-morphs. We show how to extend the assignment $\Lambda \mapsto C^*(\Lambda )$ to a functor from this category to the category whose objects are $C^*$-algebras and whose morphisms are isomorphism classes of $C^*$-correspondences.References
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Bibliographic Information
- Alex Kumjian
- Affiliation: Department of Mathematics (084), University of Nevada, Reno, Nevada 89557-0084
- Email: alex@unr.edu
- David Pask
- Affiliation: School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia
- Email: dpask@uow.edu.au
- Aidan Sims
- Affiliation: School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia
- MR Author ID: 671497
- Email: asims@uow.edu.au
- Received by editor(s): December 6, 2007
- Received by editor(s) in revised form: June 30, 2009
- Published electronically: December 20, 2010
- Additional Notes: This research was supported by the Australian Research Council.
- © Copyright 2010 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 363 (2011), 2599-2626
- MSC (2000): Primary 46L05
- DOI: https://doi.org/10.1090/S0002-9947-2010-05152-9
- MathSciNet review: 2763728