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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Generalised morphisms of $ k$-graphs: $ k$-morphs


Authors: Alex Kumjian, David Pask and Aidan Sims
Journal: Trans. Amer. Math. Soc. 363 (2011), 2599-2626
MSC (2000): Primary 46L05
Published electronically: December 20, 2010
MathSciNet review: 2763728
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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.


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Additional 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
Email: asims@uow.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05152-9
PII: S 0002-9947(2010)05152-9
Keywords: $C^{*}$-algebra, graph algebra, $k$-graph, $C^{*}$-correspondence.
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.
Article copyright: © Copyright 2010 American Mathematical Society