  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.
