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arXiv · 1208.2049

Remark on Serre $C^*$-algebras

Igor Nikolaev
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Abstract

We study non-commutative algebraic geometry of Artin, Serre and Tate in terms of the operator algebras. Namely, we define the Serre $C^*$-algebra $\mathcal{A}_X$ of a projective variety $X$ as the norm-closure of a representation of the twisted homogeneous coordinate ring of $X$ by the linear operators on a Hilbert space $\mathcal{H}$. It is proved that $X$ is homeomorphic to the space of all irreducible representations of the crossed product of $\mathcal{A}_X$ by an automorphism of $\mathcal{A}_X$. The case of rational elliptic curves $X$ is considered in detail.

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Igor Nikolaev. 2012-08-09. Remark on Serre $C^*$-algebras. https://arxiv.org/abs/1208.2049

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