fa.functional analysis - What is the smallest $C^*$-algebra containing the "standard" pseudodifferential operators?

Is $Psi^0(mathbb{R})$ (pseudodifferential operators with symbols obeying
$
|partial^alpha_x partial^beta_xi a(x,xi)| leq C_{alpha,beta} (1+|xi|)^{-|beta|}
$
) a $C^*$-algebra?

In other words, is $Psi^0(mathbb{R})$ is closed in $mathcal{L}(L^2(mathbb{R}))$ in the operator norm topology?

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If not, then is there any nice characterization by the $C^*$-algebra generated by $Psi^0$? Alternatively, what is the strongest (or just a reasonable) topology on $mathcal{L}(L^2(mathbb{R}))$ such that $Psi^0$ is a closed subspace?

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Edit: Per Yemon Choi's comments below, the above question seems somewhat hopeless. As described here, $Psi^0(mathbb{R})$ is a Fréchet $*$-algebra with a topology stronger than the operator topology. I assume that this is the topology given by the seminorms on symbols:
$$
Vert a Vert_{alpha,beta} = sup_{x,xi in mathbb{R}} (1+|xi|)^{|beta|} |partial^alpha_x partial^beta_xi a(x,xi)|.
$$

So, in addition to the above question, I am adding the following question, to make it so that there might be an answer:

Is there a reasonable description of the smallest $C^*$-algebra containing $Psi^0$?