Remark on Serre $C^{\ast }$-algebras

Abstract

We study non-commutative algebraic geometry of Artin, Serre and Tate in terms of the operator algebras. Namely, we define the Serre $C^{\ast }$-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.

9 pages; an update