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Homology for higher-rank graphs and twisted C*-algebras

Journal Article


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Abstract


  • We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a k-graph coincides with the homology of its topological realisation as described by Kaliszewski et al. We exhibit combinatorial versions of a number of standard topological constructions, and show that they are compatible, from a homological point of view, with their topological counterparts. We show how to twist the C*-algebra of a k-graph by a T-valued 2-cocycle and demonstrate that examples include all noncommutative tori. In the appendices, we construct a cubical set from a k-graph Λ and demonstrate that the homology and topological realisation of Λ coincide with those of as defined by Grandis.

Publication Date


  • 2012

Citation


  • Kumjian, A., Pask, D. & Sims, A. (2012). Homology for higher-rank graphs and twisted C*-algebras. Journal of Functional Analysis, 263 (6), 1539-1574.

Scopus Eid


  • 2-s2.0-84864038434

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=9232&context=infopapers

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/1897

Number Of Pages


  • 35

Start Page


  • 1539

End Page


  • 1574

Volume


  • 263

Issue


  • 6

Place Of Publication


  • http://www.sciencedirect.com/science/article/pii/S0022123612002224

Abstract


  • We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a k-graph coincides with the homology of its topological realisation as described by Kaliszewski et al. We exhibit combinatorial versions of a number of standard topological constructions, and show that they are compatible, from a homological point of view, with their topological counterparts. We show how to twist the C*-algebra of a k-graph by a T-valued 2-cocycle and demonstrate that examples include all noncommutative tori. In the appendices, we construct a cubical set from a k-graph Λ and demonstrate that the homology and topological realisation of Λ coincide with those of as defined by Grandis.

Publication Date


  • 2012

Citation


  • Kumjian, A., Pask, D. & Sims, A. (2012). Homology for higher-rank graphs and twisted C*-algebras. Journal of Functional Analysis, 263 (6), 1539-1574.

Scopus Eid


  • 2-s2.0-84864038434

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=9232&context=infopapers

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/1897

Number Of Pages


  • 35

Start Page


  • 1539

End Page


  • 1574

Volume


  • 263

Issue


  • 6

Place Of Publication


  • http://www.sciencedirect.com/science/article/pii/S0022123612002224