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)).

/pdf/on-the-k-theory-of-higher-rank-graph-c-algebras-4fyhxo1jp3.pdf

Infinite Abelian groups

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

Let R be a subring of the rationals. We want to investigate self splitting R-modules G that is Ext R(G; G) = 0 holds and follow Schultz [22] to call such modules splitters. Free modules and torsion-free cotorsion modules are classical examples for splitters. Are there others? Answering an open problem by Schultz [22] we will show that there are more splitters, in fact we are able to prescribe their endomorphism R-algebras with a free R-module structure. As a byproduct we are able to answer a problem of Salce [21] showing that all rational cotorsion theories have enough injectives and enough projectives.

/pdf/cotorsion-theories-and-splitters-1h958tluuo.pdf

Cotorsion theories and splitters

We consider the ordinary differential equation [Bu(t)]′ = Au(t) with A and B linear operators with domains in a Banach space X and ranges in a Banach space Y. The initial condition is that the limit as (t → 0 of Bu(t) is prescribed in Y. We study the properties of the “solution operator” S(t) which maps the initial state in Y to the solution u(t) at time t. The notion of infinitesimal generator A of S(t) is introduced and the relationships between S(t), an associated semi-group E(t), the operators A and B and some other operators are studied. In particular a pair of operators Ao and Bo, derived from A and B, determine the family S(t) of operators. These so-called “generating pairs” are characterized. The operators A and B and Ao and Bo need not be closed, but form so-called closed pairs which is a weaker condition. We also discuss two applications of the theory.

Infinite Abelian groups

Citations

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

Cotorsion theories and splitters

Linear evolution equations in two Banach spaces

Implications of large-cardinal principles in homotopical localization

The Baer-invariant of a direct product

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