On the K-theory of the AF core of a graph C*-algebra

In this paper, we study multiplicative structures on the K-theory of the core $A:=C^{\ast }(E{)}^{U(1)}$ of the C*-algebra $C^{\ast }(E)$ of a directed graph $E$. In the first part of the paper, we study embeddings $E\to E\times E$ that induce a -homomorphism $A\otimes A\to A$. Through Künneth formula, any such a -homomorphism induces a ring structure on $K_{\ast }(A)$. In the second part, we give conditions on $E$ such that $K_{\ast }(A)$ is generate by “noncommutative line bundles” (invertible bimodules). The same conditions guarantee the existence of a homomorphism of abelian groups $K_0(A)\to \mathbb{Z}[\lambda ]\mathrm{/}(\det (\lambda \mathrm{\Gamma }-1))$ (where $\mathrm{\Gamma }$ is the adjacency matrix of $E$) that is compatible with the tensor product of line bundles. Examples include the C-algebra $C(\mathbb{C}{P}_q^{n-1})$ of a quantum projective space, the $UHF(n^{\mathrm{\infty }})$ algebra, and the C-algebra of the space parameterizing Penrose tilings. For the first algebra, as a corollary we recover some identities that classically follow from the ring structure of $K^0(\mathbb{C}{P}^{n-1})$, and that were proved by Arici, Brain and Landi in the quantum case. Incidentally, we observe that the C*-algebra of Penrose tilings is the AF core of the Cuntz algebra ${\mathcal{O}}_2$, if the latter is realized using the appropriate graph.