The Laplacian of an operator and the radial Toeplitz algebra

Using tools from quantum harmonic analysis, we show that the domain of the Laplacian of an operator is dense in the Toeplitz algebra over the Fock space ${\mathcal{F}}^2({\mathbb{C}}^n)$. As an application, we provide a simplified treatment of the Gelfand theory of the radial Toeplitz algebra.