On the K-theory of higher rank graph C ∗ -algebras

By Evans, Dafydd Gwion. 2004

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Given a row-finite k-graph Λ with no sources we investigate the K-theory of the higher rank graph C ∗ -algebra, C ∗ (Λ). When k = 2 we are able to give explicit formulae to calculate the K-groups of C ∗ (Λ). The K-groups of C ∗ (Λ) for k> 2 can be calculated under certain circumstances and we consider the case k =3 . We prove that for arbitrary k, the torsion-free rank of K0(C ∗ (Λ)) and K1(C ∗ (Λ)) are equal when C ∗ (Λ) is unital, and for k = 2 we determine the position of the class of the unit of C ∗ (Λ) in K0(C ∗ (Λ)).

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/81d9265cdd2b23a313ac238f2392bcbfac9e88ed

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In Algebra and Number Theory

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Citations in the corpus

Ample groupoids: equivalence, homology, and Matui's HK conjecture (2018, methodology): “So Evans’ spectral sequence collapses, and we obtain K∗(C ∗(Λ)) = 0 as well.”
Remarks on some fundamental results about higher-rank graphs and their C*-algebras (2011, background): “C∗-algebras associated to graphs and k-graphs have attracted significant attention recently because they at once encompass a great many interesting examples [2, 6, 11, 17], and are remarkably tractable [3, 4, 5, 7, 12, 13, 21].”
Homology for higher-rank graphs and twisted C*-algebras (2011, background): “The C∗-algebras of higher-rank graphs have been studied by numerous authors over the last decade (see, for example, [6, 7, 10, 11, 40, 41, 43]).”
Rank-two graphs whose C∗-algebras are direct limits of circle algebras (2005, methodology): “ diagrams. In Theorem 4.3 we compute the K-theory of C∗(Λ) for a rank-2 Bratteli diagram Λ. Our arguments are elementary and do not depend on the computations of K-theory for general 2-graph algebras [29, 12]. The K-theory calculation shows in particular that K1(C∗(Λ)) is isomorphic to a subgroup Go”