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

Question

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 $\Psi^0$?

Answer 1

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.

Answer 2

A remark to add to Paul's answer: yes $\mathcal{B}(U)$ is precisely the $C^*$-algebra generated by $\Psi^0$, as follows.

Let $\mathcal{A}$ be the $C^\ast$-closure of $\Psi^0$. The image of the principal symbol map $\mathrm{Symb}\colon \Psi^0 \to C_0(S^\ast U)$ is dense, and so the image of $\mathcal{A}$ is all of $C_0(S^\ast U)$. Thus, for any $T\in\mathcal{B}(U)$, we can find $T^\prime \in \mathcal{A}$ with the same symbol. Following Paul, $T-T^\prime$ is a compact operator. The compacts are in $\mathcal{A}$ (since the smoothing operators are dense therein). Thus, $T$ is in $\mathcal{A}$.

edit (AlexE): To phrase this result in another way, we have the so-called pseudodifferential operator extension (a short exact sequence) $$0 \to \mathcal{K} \to \mathcal{A} \stackrel{\mathrm{Symb}}\longrightarrow C_0(S^\ast U) \to 0,$$ where $\mathcal{K}$ are the compact operators.