Isomorphisms of reduced group $C^*$-algebras with unique trace lift to the group von Neumann algebras. Anyone have a nice proof or reference of the following fact. Suppose, G,H are infinite discrete groups and $\theta: C^*_r(G) \to C^*_r(H)$ is a $*$-isomorphism and $C^*_r(G)$ has a unique trace then $\theta$ lifts to an isomorphism of $L(G) \simeq L(H)$. 
Note: I kind of want to say this was proven in a Kadison/Ringrose paper but cannot remember the reference.
 A: As $C_r^*(G)$ has unique trace, such trace $\tau$ is the canonical trace $\tau(x)=\langle x\delta_e,\delta_e\rangle$, so in particular it is faithful and normal. And the isomorphism guarantees that the same is true for $C_r^*(H)$. You also have that $\tau\circ\theta=\tau$ by the uniqueness. 
Fix $x\in L(G)$. Then there is $\{x_j\}\subset C_r^*(G)$ and $x_j\xrightarrow{\text{wot}} x$. This means that 
$$
\langle x_j\delta_y,\delta_z\rangle\to \langle x\delta_y,\delta_z\rangle,\ \ \ y,z\in G.
$$
Define a sequilinear form on $\ell^2(H)$ by 
$$
[\delta_y,\delta_z]=\langle x\,\delta_{\theta^{-1}(y)},\delta_{\theta^{-1}(z)}\rangle,\ \ \ y,z\in \ell^2(H)
$$
and extend to arbitrary elements of $\ell^2(H)$. By Riesz Representation Theorem there exists $\theta(x)\in B(\ell^2(H))$ with 
$$\tag1
\langle \theta(x)\delta_y,\delta_z\rangle = \langle x\,\delta_{\theta^{-1}(y)},\delta_{\theta^{-1}(z)}\rangle,\ \ \ y,z\in H.
$$
For any $T\in L(H)'$, 
\begin{align}
\langle T\theta(x)\delta_y,\delta_z\rangle&=\langle \theta(x)\delta_y,T^*\delta_z\rangle
=\lim_j \langle \theta(x_j)\delta_y,T^*\delta_z\rangle
=\lim_j \langle T\theta(x_j)\delta_y,\delta_z\rangle\\ \ \\
&=\lim_j \langle \theta(x_j)T\delta_y,\delta_z\rangle
= \langle \theta(x)T\delta_y,\delta_z\rangle.
\end{align}
As the above works for all $y,z$ we get that $\theta(x)\in L(H)''=L(H)$. It is easy to check from $(1)$ that $\theta$ is a $*$ homomorphism, and also that $\theta$ as defined above restricts to the old $\theta$ on $C^*_r(G)$, and it still satisfies $\tau\circ\theta=\tau$, now over all of $L(G)$.  From this and the faithfullness of $\tau$ is follows trivially that $\theta$ is injective: if $\theta(x)=0$, then $\theta(x^*x)=\theta(x)^*\theta(x)=0$, and so $=\tau(\theta(x^*x))=\tau(x^*x)$, so $x=0$. 
It remains to see that $\theta$ is surjective. If $x'\in L(H)$, there exists a net $\{x_j'\}\subset C^*_r(H)$ with $x_j'\xrightarrow{\text{wot}} x'$. Let $x_j\in C^*_r(G)$ with $\theta(x_j)=x_j'$. Again the equality $(1)$ shows that $\{\theta(x_j)\}$ is wot-convergent in $L(G)$, and if $x=\lim x_j$ then $\theta(x)=x'$. 
