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](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=MR&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=MR90d%3A47016&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq), $\Psi^0(\mathbb{R})$ 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 $\Psi^0$? 



What is the smallest $C^*$-algebra containing the "standard" pseudodifferential operators?
I have to confess to being more confused by the theory of pseudodifferential operators than I should be, but I think an answer to a question at least related to yours is briefly discussed in chapter 2 of Higson and Roe's Analytic K-homology.


Begin with an open subset $U$ of $\mathbb{R}^n$ and consider a smooth complex valued function $\sigma$ on $T^\*(U)$ with the following properties:


* $\sigma(x, t \xi) = \sigma(x, \xi)$ for $t \geq 1$, $|\xi| \geq 1$
* $\sigma(x, \xi)$ vanishes for $x$ outside of some compact subset of $U$


Define the pseudodifferential operator associated to $\sigma$ to be the operator


$D\_\sigma f(x) = \frac{1}{(2 \pi)^n} \int \sigma(x, \xi) \hat{f}(\xi) e^{i(x,\xi)} d\xi$


The first condition on $\sigma$ above implies that $\sigma$ gives rise to a function on the cosphere bundle $S^\*M$ (the symbol of $D\_\sigma$). $D\_\sigma$ extends to a bounded operator on $L^2(U)$, so consider the C\*-algebra $\mathcal{B}(U)$ generated by the $D\_\sigma$. Higson and Roe assert that the map sending $D\_\sigma$ to its symbol extends to a surjective $\*$-homomorphism from $\mathcal{B}(U)$ to $C\_0(S^\* U)$ whose kernel is precisely the C\* algebra of compact operators on $L^2(U)$.


Thus a certain class of pseudodifferential operators generates a C\* algebra extension of $C\_0(U)$ by the compact operators. I make no claim about the relationship between $\mathcal{B}(U)$ and your $\Psi^0$, but maybe this statement is still useful to you.


