Formal Deformation quantization as a Fréchet algebra

Abstract

We define a Fréchet topology on the space $C^{\infty }(X)[[\hslash ]]$ of formal smooth functions on a symplectic manifold $X$, by constructing a sequence of semi-norms on it. For any star product $\star$ on $C^{\infty }(X)[[\hslash ]]$ making it a formal deformation quantization of $X$, we will show that the quantum product $\star$ is jointly continuous, and making it a Fréchet 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 Fréchet topology.