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Graphs of C*-correspondences and Fell bundles

Journal Article


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Abstract


  • We define the notion of a $\Lambda$-system of $C^*$-correspondences associated to a higher-rank graph $\Lambda$. Roughly speaking, such a system assigns to each vertex of $\Lambda$ a $C^*$-algebra, and to each path in $\Lambda$ a $C^*$-correspondence in a way which carries compositions of paths to balanced tensor products of $C^*$-correspondences. Under some simplifying assumptions, we use Fowler's technology of Cuntz-Pimsner algebras for product systems of $C^*$-correspondences to associate a $C^*$-algebra to each $\Lambda$-system. We then construct a Fell bundle over the path groupoid $\Gg_\Lambda$ and show that the $C^*$-algebra of the $\Lambda$-system coincides with the reduced cross-sectional algebra of the Fell bundle. We conclude by discussing several examples of our construction arising in the literature.

Publication Date


  • 2010

Citation


  • Deaconu, V., Kumjian, A., Pask, D. & Sims, A. (2010). Graphs of C*-correspondences and Fell bundles. Indiana University Mathematics Journal, 59 (5), 1687-1736.

Scopus Eid


  • 2-s2.0-80053397545

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=10970&context=infopapers&unstamped=1

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/3634

Number Of Pages


  • 49

Start Page


  • 1687

End Page


  • 1736

Volume


  • 59

Issue


  • 5

Place Of Publication


  • http://www.iumj.indiana.edu/IUMJ/FULLTEXT/2010/59/3893

Abstract


  • We define the notion of a $\Lambda$-system of $C^*$-correspondences associated to a higher-rank graph $\Lambda$. Roughly speaking, such a system assigns to each vertex of $\Lambda$ a $C^*$-algebra, and to each path in $\Lambda$ a $C^*$-correspondence in a way which carries compositions of paths to balanced tensor products of $C^*$-correspondences. Under some simplifying assumptions, we use Fowler's technology of Cuntz-Pimsner algebras for product systems of $C^*$-correspondences to associate a $C^*$-algebra to each $\Lambda$-system. We then construct a Fell bundle over the path groupoid $\Gg_\Lambda$ and show that the $C^*$-algebra of the $\Lambda$-system coincides with the reduced cross-sectional algebra of the Fell bundle. We conclude by discussing several examples of our construction arising in the literature.

Publication Date


  • 2010

Citation


  • Deaconu, V., Kumjian, A., Pask, D. & Sims, A. (2010). Graphs of C*-correspondences and Fell bundles. Indiana University Mathematics Journal, 59 (5), 1687-1736.

Scopus Eid


  • 2-s2.0-80053397545

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=10970&context=infopapers&unstamped=1

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/3634

Number Of Pages


  • 49

Start Page


  • 1687

End Page


  • 1736

Volume


  • 59

Issue


  • 5

Place Of Publication


  • http://www.iumj.indiana.edu/IUMJ/FULLTEXT/2010/59/3893