Proving Uniqueness of CCR Algebras I am currently trying to learn about CCR algebras (canonical commutation relations) and I am experiencing some confusion with the proof that the CCR algebra of a (non-degenerate, real) symplectic vector space is unique.
I have been reading from chapter 6 of the following, in which the relevant result is Theorem 3:
https://www.math.uni-potsdam.de/fileadmin/user_upload/Prof-Geometrie/Dokumente/Publikationen/qft-alg.pdf
I shall summarise the proof, and insert numbers where I have questions. Any terminology should be defined in the document linked above.
Proof:
Given two CCR-representations ($A_1$, $W_1$) and ($A_2$, $W_2$) of the symplectic vector space ($V$, $\omega$), we must show that the $^*$-isomorphism $\pi:\langle W_1(V)\rangle\rightarrow\langle W_2(V)\rangle$ between the $^*$-algebras $\langle W_1(V)\rangle$ and $\langle W_2(V)\rangle$ extends to an isometry between $A_1$ and $A_2$$^{(1)}$. 
Then the norm $\|x\|=\|\pi(x)\|_2$ is introduced on $A_1$, and it noted that $\|\pi(x)\|_2\leq \|x\|_\text{max}$, where $\|\cdot\|_\text{max}$ is defined in Lemma 9 by 
$$\|x\|_\text{max}=\sup\{\|x\|\text{; }\|\cdot\|\text{ is a C}^*\text{ norm on } \langle W_1(V)\rangle\}.$$
Then it is concluded that $\varphi$ extends to a $^*$-homomorphism $\overline{\langle W_1(V)\rangle}^\text{max}\rightarrow A_2$$^{(2)}$. Lemma 10 is applied to conclude that this extension is injective, and then it follows that the extension is isometric$^{(3)}$.
This is pretty much where the proof finishes, barring one note about the case when $A_1=A_2$. I'll now list my questions.
(1) Checking this will be sufficient because we could then do the same for the inverse $\varphi^{-1}$ to get the required map between $A_1$ and $A_2$? 
(2) The bound $\|\pi(x)\|_2\leq \|x\|_\text{max}$ shows that $\pi$ is continuous, so $\pi$ extends to $\overline{\langle W_1(V)\rangle}^\text{max}$ by continuity and density of $\langle W_1(V)\rangle$?
(3) I really don't understand why this is sufficient. To me it seems that finishing at this point requires the completion $\overline{\langle W_1(V)\rangle}^\text{max}$ to coincide with $A_1$, but I haven't really gotten anywhere with showing this on my own.
Any help/suggestions/slaps around the face because it's obvious will be greatly appreciated. 
 A: I've had a good old think about this, and may have filled in some of the blanks. 
For starters I have some additional confusion in the author's definition of CCR-algebras, they say that if $(A,W)$ is a Weyl system of $(V,\omega)$ then $A$ is a CCR-algebra if $A$ is generated as a C$^*$-algebra by the set $\{W(u)\text{; }u\in V\}$. All texts I have encountered only define C$^*$-subalgebras generated by subsets so I assume that what is meant in this case is that $A$ coincides with the C$^*$-subalgebra of itself generated by $\{W(u)\text{; }u\in V\}$. If this is what is meant, then it is possible to show using the properties of a Weyl system that $A$ is the norm-closure of the linear span of the set $\{W(u)\text{; }u\in V\}$. 
Consider the $^*$-isomorphism $\pi:\langle W_1(V) \rangle\rightarrow\langle W_2(V) \rangle$. The set $\langle W_2(V) \rangle$ is dense in $A_2$, and the image of the composition
$$\langle W_1(V) \rangle \overset{\pi}{\rightarrow}\langle W_2(V) \rangle \overset{i}{\hookrightarrow} A_2,$$
(where $i$ denotes inclusion), is $\langle W_2(V) \rangle$. (Perhaps it is worth noting that this composition is a $*$-homomorphism). 
Defining $\|x\|:=\|\pi(x)\|_2$ gives a C$^*$-norm on $\langle W_1(V) \rangle$ so there is the bound $\|\pi(x)\|_2=\|x\|\leq \|x\|_\text{max}$ which shows that the composition above is continuous and we may extend it to a map $\varphi:\overline{\langle W_1(V) \rangle}^\text{max}\rightarrow A_2$. Now we invoke Lemma 10 to deduce that $\varphi$ is injective and hence isometric.
Since $\varphi$ extends $i\circ \pi$, the image of $\varphi$ must contain $\langle W_2(V) \rangle$, but since $\varphi$ is an isometry into a complete metric space, its image must be closed, therefore $\varphi$ is surjective, so is an isomorphism. (Presumably the extension $\varphi$ is also a $^*$-homomorphism too).
I am still a bit stuck on the last part.

I think that I'm going to take a break from thinking about this, and return in a day or two.
