Skip to main content
placeholder image

Real rank and topological dimenstion of higher rank graph algebras

Journal Article


Abstract


  • We study dimension theory for the C*-algebras of row-finite k-graphs with

    no sources. We establish that strong aperiodicity—the higher-rank analogue of condition

    (K)—for a k-graph is necessary and sufficient for the associated C*-algebra to have

    topological dimension zero. We prove that a purely infinite 2-graph algebra has real-rank

    zero if and only if it has topological dimension zero and satisfies a homological condition

    that can be characterised in terms of the adjacency matrices of the 2-graph. We also

    show that a k-graph C*-algebra with topological dimension zero is purely infinite if and

    only if all the vertex projections are properly infinite. We show by example that there are

    strongly purely infinite 2-graphs algebras, both with and without topological dimension

    zero, that fail to have real-rank zero.

Publication Date


  • 2017

Citation


  • Pask, D., Sierakowski, A. & Sims, A. (2017). Real rank and topological dimenstion of higher rank graph algebras. Indiana University Mathematics Journal, 66 (6), 2137-2168.

Number Of Pages


  • 31

Start Page


  • 2137

End Page


  • 2168

Volume


  • 66

Issue


  • 6

Place Of Publication


  • United States

Abstract


  • We study dimension theory for the C*-algebras of row-finite k-graphs with

    no sources. We establish that strong aperiodicity—the higher-rank analogue of condition

    (K)—for a k-graph is necessary and sufficient for the associated C*-algebra to have

    topological dimension zero. We prove that a purely infinite 2-graph algebra has real-rank

    zero if and only if it has topological dimension zero and satisfies a homological condition

    that can be characterised in terms of the adjacency matrices of the 2-graph. We also

    show that a k-graph C*-algebra with topological dimension zero is purely infinite if and

    only if all the vertex projections are properly infinite. We show by example that there are

    strongly purely infinite 2-graphs algebras, both with and without topological dimension

    zero, that fail to have real-rank zero.

Publication Date


  • 2017

Citation


  • Pask, D., Sierakowski, A. & Sims, A. (2017). Real rank and topological dimenstion of higher rank graph algebras. Indiana University Mathematics Journal, 66 (6), 2137-2168.

Number Of Pages


  • 31

Start Page


  • 2137

End Page


  • 2168

Volume


  • 66

Issue


  • 6

Place Of Publication


  • United States