properly supported, then it actually defines a linear map $C^\infty_c({\mathbb{R}}^n) \to C^\infty_c({\mathbb{R}}^n)$. We denote by $\Psi_H^m({\mathbb{R}}^n)$ the vector space of such properly-supported operators and by $\Psi_{H,c}^m({\mathbb{R}}^n)$ its subspace of *compactly* supported operators. Since properly-supported operators can be composed, the unions of all-orders operators $$\Psi_H({\mathbb{R}}^n) = \bigcup_{m \in {\mathbb{R}}} \Psi_H^m({\mathbb{R}}^n)\ ,\qquad  \Psi_{H,c}({\mathbb{R}}^n) = \bigcup_{m \in {\mathbb{R}}} \Psi_{H,c}^m({\mathbb{R}}^n)$$ are associative algebras over ${\mathbb{C}}$. The ideals of *regularizing operators* $$\Psi^{-\infty}({\mathbb{R}}^n) = \bigcap_{m \in {\mathbb{R}}} \Psi_H^m({\mathbb{R}}^n) \ ,\qquad         \Psi^{-\infty}_c({\mathbb{R}}^n) = \bigcap_{m \in {\mathbb{R}}} \Psi_{H,c}^m({\mathbb{R}}^n)$$ correspond respectively to the algebras of operators with properly- and compactly-supported smooth Schwartz kernel.\
If $P_1,P_2 \in \Psi_H({\mathbb{R}}^n)$ are Heisenberg pseudodifferential operators of symbols $\sigma_1$ and $\sigma_2$, $P_1P_2$ is a Heisenberg pseudodifferential operator whose symbol $\sigma$ is given by the *star-product* of symbols : $$\label{star product} 
\sigma(x, p) = \sigma_1 \star \sigma_2 (x,p) \sim \sum_{\vert \alpha \vert \geq 0} \frac{(-{\textup{\textbf{i}}})^{\vert \alpha \vert}}{\alpha!} \partial^\alpha_p \sigma_1(x, p) \partial^\alpha_x \sigma_2(x, p)$$ Note that the order of each symbol in the sum is decreasing while $\vert \alpha \vert$ is increasing.\
We define the *algebra of Heisenberg formal classical symbols* ${\mathcal{S}}_H({\mathbb{R}}^n)$ and its compactly-supported subalgebra ${\mathcal{S}}_{H,c}({\mathbb{R}}^n)$ as quotients $${\mathcal{S}}_H({\mathbb{R}}^n) = \Psi_H({\mathbb{R}}^n)/\Psi^{-\infty}({\mathbb{R}}^n) \ ,\qquad {\mathcal{S}}_{H,c}({\mathbb{R}}^n) = \Psi_{H,c}({\mathbb{R}}^n)/\Psi_c^{-\infty}({\mathbb{R}}^n)$$ Their elements are formal sums given in (\[classical hpdo\]), 