Abstract

We study multiplicative structures on the K-theory of the core $A$ of the C*-algebra of a directed graph $E$. We first study embeddings $E\to E\times E$ that induce a *-homomorphism $A\otimes A\to A$. Through the Künneth formula, any such *-homomorphism induces a ring structure on $K_{\ast }(A)$. We then give conditions on $E$ for which $K_{\ast }(A)$ is generated by “noncommutative line bundles” (invertible bimodules). The same conditions guarantee the existence of a homomorphism of abelian groups $K_0(A)\to \mathbb{Z}[\lambda ]∕(\det <!--FUNCTION\; APPLICATION-->(\lambda \Gamma -1))$ (where $\Gamma$ is the adjacency matrix of $E$) that is compatible with the tensor product of line bundles. Examples include the C*-algebra $C(ℂP_q^{n-1})$ of a quantum projective space, the $\mathrm{UHF}<!--FUNCTION\; APPLICATION-->(n^{\infty })$ algebra, and the C*-algebra of the space parametrizing Penrose tilings. For the first algebra, we recover as a corollary some identities that classically follow from the ring structure of $K^0(ℂP^{n-1})$ 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.

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