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I am reading from the following document, and am a bit stumped by footnote 4 on page 5:

https://arxiv.org/pdf/math-ph/0006011.pdf

Actually, I will copy the relevant text because it disappears off the bottom of the page (or at least it does when I view it on my laptop).

"The fact that the CCR-algebra is simple can be seen as the correct generalization of the Stone-von Neumann uniqueness theorem to the case of infinitely many degrees of freedom. Indeed, since $\mathcal{A}(H,\sigma)$ is simple, all of its representations are isomorphic. When H is finite-dimensional, this isomorphism is unitarily implementable, entailing the result in [20]."

The object $\mathcal{A}(H,\sigma)$ which is mentioned is a C$^*$-algebra.

I have found only one mention of isomorphic representations for C$^*$-algebras, which is Definition 2 in the following document:

http://www.math.ru.nl/~tcrisp/teaching/2017-Cstar-reps/notes/2017-09-25-Cstar-reps-notes.pdf

And my problem is that I can't make any progress with proving the implication hinted at by the footnote, that if $A$ is a C$^*$-algebra, then

$$\text{A simple }\Rightarrow \text{ All of it's representations are isomorphic.}$$

Well, to me this is what the footnote is hinting at, but it could well be the case that the author just means that this is the case for $\mathcal{A}(H,\sigma)$, so I guess I should also be thinking about possible counter examples for the general case. However I'd be the first to admit that my knowledge of representation theory isn't particularly great yet. . .

As usual, I would appreciate any hints/suggestions e.t.c. I'm really quite interested by this footnote, as it seems to say that Slawny's Theorem implies the Stone-Von Neumann Theorem which is something I've not seen anywhere else.

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It is not true that simplicity implies isomorphism of representations (it may be true for your algebra, that I don't know what it is).

I cannot immediately think of an elementary example, but here are a couple.

  • Take the free groups $\mathbb F_2$ and $\mathbb F_3$ and their reduced C$^*$-algebras. The canonical homomorphism $\mathbb F_3\to\mathbb F_2$ lifts to a $*$-epimomorphism $C_r^*(\mathbb F_3)\to C_r^*(\mathbb F_2)$. Compare with the identity representation $C_r^*(\mathbb F_3)\to C_r^*(\mathbb F_3)$ and the fact that $C_r^*(\mathbb F_3)$ and $C_r^*(\mathbb F_2)$ are not isomorphic (this is due to Pimsner-Voiculescu if I'm not wrong), and that both are simple.

  • Take the Cuntz algebras $\mathbb O_2$ and $\mathbb O_3$. It is well-known that they are not isomorphic, and $\mathbb O_3$ embeds in $\mathbb O_2$ (all exact algebras embed in it). So again we get two representations of $\mathbb O_3$ (one into itself, one into $\mathbb O_2$) that are not isomorphic. And Cuntz algebras are simple.

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