Remark on Serre $C^*$-algebras

TL;DR

This paper explores the non-commutative geometric structure of projective varieties through Serre $C^*$-algebras, linking algebraic and operator algebra perspectives, with detailed analysis for rational elliptic curves.

Contribution

It introduces the Serre $C^*$-algebra for projective varieties and establishes a homeomorphism between the variety and the space of irreducible representations of a related crossed product.

Findings

X is homeomorphic to the space of irreducible representations of the crossed product of $\

Detailed analysis of rational elliptic curves within this framework.

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.