The screenshot is from Haagerup's paper "uniqueness of the injective III$_1$-factor".
I have two questions :
- If $N$ is properly infinite, how to show that $N\otimes M_{2}(\Bbb C)\cong N$?
2.Why we can choose a $*$-isomorphism $\Phi：N\otimes M_{2}(\Bbb C)\rightarrow N$ such that .....?
By Connes-stormer transitivity, we only know the following fact:
For any two normal states $\phi$ and $\psi$ on a von Neumann algebra $M$, take an arbitrary $\epsilon＞0$, there exists a unitary $u\in M$ such that $\|\phi- u\psi u^*\|<\epsilon$. I don't understand how to construct $\Phi$ which is mentioned in the proof.
Because $N$ is properly infinite, you can halve the identity: there exists a projection $p\in N$ such that $1\sim p\sim 1-p$. Let $v,w\in N$ with $$v^*v=1, \quad vv^*=p,\quad w^*w=p, \quad ww^*=1-p.$$ Now we can define $\tilde\gamma:N\to M_2(N)$ by $$ \tilde\gamma(x)=\begin{bmatrix}v^*xv & v^*xw\\ w^*xv& wv^*xvw^* \end{bmatrix}. $$ It is straightforward to check that this is a $*$-homomorphism. Injectivity is easy because if $\tilde\gamma(x)=0$ then $0=pxp=px(1-p)=(1-p)xp=(1-p)x(1-p)$ and so $x=0$. For surjectivity, $$ \begin{bmatrix} a& b\\ c&d\end{bmatrix} =\tilde\gamma\big(vav^*+vbw^*+wcv^*+wdw^*\big). $$
Now you apply Connes-Størmer transivity to $\varepsilon=\tfrac12$ and the states $\psi$ and $(\psi\otimes \omega_1)\circ\tilde\gamma^{-1}$: there exists an (inner) automorphism $\theta$ of $N$ such that $$ \|(\psi\otimes\omega_1)\circ\tilde\gamma^{-1}\circ\theta-\psi\|<\tfrac12. $$ Finally, name $\Phi=\theta^{-1}\circ\tilde\gamma$.