$K$-Theory and $K$-Homology of $C^*$-Algebras for Row-Finite Graphs

Fredholm Modules Over Graph -Algebras

“…which we view as a chain complex A * (G) concentrated in degrees 1 and 0. Theorem 2.1 [6,8,17,20,26]. The chain complex A * (G) computes the K-theory of C * (G):…”

“…The latter requirement was removed in [8]. The assumption of Condition (L) was avoided in [26], using a method quite different from that of Cuntz and Krieger, although the argument would appear to be incomplete as regards odd K-homology (the proof of [26,Lemma 3.11] asserts that K 1 (C * (E)) embeds in Hom(K 1 (C * (E)), Z), which is true only if K 1 (C * (E)) is torsion-free).…”

“…The latter requirement was removed in [DT02]. The assumption of Condition (L) was avoided in [Yi07], using an argument quite different to that of Cuntz and Krieger. In Section 1 we present the formulas deriving from [CK80], [Cun81] and [Tom03]. The only novelty of our presentation is the translation from the language of extensions to that of Fredholm modules, and the removal of the assumption of Condition (L).…”

Graded $K$-theory and $K$-homology of relative Cuntz-Pimsner algebras and graph $C^*$-algebras

“…Cuntz proved that the K-theory groups of the C * -algebra of a finite directed graph E with no sources and with {0, 1}-valued adjacency matrix A E are the cokernel and kernel of the matrix 1 − A t E regarded as an endomorphism of the free abelian group ZE 0 (see [7,Proposition 3.1]). This was generalised to row-finite directed graphs E with no sources in [24], and later (with appropriate adjustments made to the domain and the codomain of A t E in each case) to all row-finite directed graphs E in [28], and to arbitrary graphs in [1,10] (see also [13,27,32,33,36]).…”

“…It is less of an automatic reaction to compute K-homology for C * -algebras, but, for example, Cuntz and Krieger computed (in [6,Theorem 5.3]) the Ext-group (that is, the odd K-homology group) of the Cuntz-Krieger algebra O A of A ∈ M n (Z + ) as the cokernel of 1 − A regarded as an endomorphism of Z n . The computation was later generalised to graph C * -algebras in [10,35] (see also [5,36]).…”