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 $A:=C^*(E)^{U(1)}$ of the C*-algebra $C^*(E)$ 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 $K_*(A)$. In the second part, we give conditions on $E$ such that $K_*(A)$ is generate by "noncommutative line bundles" (invertible bimodules). The same conditions guarantee the existence of a homomorphism of abelian groups $K_0(A)\to\mathbb{Z}[λ]/(\det(λΓ-1))$ (where $Γ$ is the adjacency matrix of $E$) that is compatible with the tensor product of line bundles. Examples include the C*-algebra $C(\mathbb{C}P^{n-1}_q)$ of a quantum projective space, the $UHF(n^\infty)$ 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.