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Generalised morphisms of k-graphs: k-morphs

Journal Article


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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.

Publication Date


  • 2011

Citation


  • Kumjian, A., Pask, D. & Sims, A. (2011). Generalised morphisms of k-graphs: k-morphs. Transactions of the American Mathematical Society, 363 (5), 2599-2626.

Scopus Eid


  • 2-s2.0-79951909147

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=9947&context=infopapers

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/2611

Has Global Citation Frequency


Number Of Pages


  • 27

Start Page


  • 2599

End Page


  • 2626

Volume


  • 363

Issue


  • 5

Place Of Publication


  • United States

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.

Publication Date


  • 2011

Citation


  • Kumjian, A., Pask, D. & Sims, A. (2011). Generalised morphisms of k-graphs: k-morphs. Transactions of the American Mathematical Society, 363 (5), 2599-2626.

Scopus Eid


  • 2-s2.0-79951909147

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=9947&context=infopapers

Ro Metadata Url


  • http://ro.uow.edu.au/infopapers/2611

Has Global Citation Frequency


Number Of Pages


  • 27

Start Page


  • 2599

End Page


  • 2626

Volume


  • 363

Issue


  • 5

Place Of Publication


  • United States