On the K-theory of the AF core of a graph C*-algebra
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911592371716096 |
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| author | D'Andrea, Francesco |
| author_facet | D'Andrea, Francesco |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_06242 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the K-theory of the AF core of a graph C*-algebra D'Andrea, Francesco K-Theory and Homology Primary: 46L80, Secondary: 46L67, 46L85 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. |
| title | On the K-theory of the AF core of a graph C*-algebra |
| topic | K-Theory and Homology Primary: 46L80, Secondary: 46L67, 46L85 |
| url | https://arxiv.org/abs/2410.06242 |