Formal Deformation quantization as a Fr\'echet algebra

TL;DR

This paper introduces a Fréchet topology on formal smooth functions on a symplectic manifold, demonstrating that star products are continuous and form a Fréchet algebra, with applications to quantum density and trace continuity.

Contribution

It constructs a Fréchet topology making deformation quantization star products jointly continuous and proves a quantum Weierstrass theorem and trace continuity.

Findings

Star products are jointly continuous in the Fréchet topology.

Quantum polynomials are locally dense among formal smooth functions.

The canonical trace is continuous under the Fréchet topology.

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.