Formal Deformation quantization as a Fr\'echet algebra

Abstract

We define a Fr\'echet topology on the space $C^\infty(X)[[\hbar]]$ of formal smooth functions on a symplectic manifold $X$, by constructing a sequence of semi-norms on it. For any star product $\star$ on $C^\infty(X)[[\hbar]]$ making it a formal deformation quantization of $X$, we will show that the quantum product $\star$ is jointly continuous, and making it a Fr\'echet algebra. We will show a quantum Weierstrass theorem which says quantum polynomials are locally dense in all formal smooth functions. We will also show that the canonical trace of any formal deformation quantization is continuous under this Fr\'echet topology.