in this post I would like to discuss the fact that If $A$ is a $C^*$-algebra, then $M_n(A^{**})\cong M_n(A)^{**}$, as mentioned in Brown and Ozawa. I can't really see it. Actually, it is enough for me to find a bijective linear map that respects positivity in both directions, so this is what I'm trying (I define the multiplication and involution on the double dual through Sherman-Takeda and not with Arens products, so it's hard for me to check if these operations are being preserved).
I recently studied the Sherman-Takeda theorem that states that if $A$ is a $C^*$-algebra, then $A^{**}$ is isometrically isomorphic to $\pi_u(A)''$, where $(H_u,\pi_u)$ is the universal representation of $A$. It is also important to know how this is done: we define the map $\sigma_*:A^*\to(\pi_u(A)'')_*$ first on $S(A)$ and then extending linearly to $\pi_u(A)''$: if $\omega\in S(A)$ then there exists a unique normal state $\bar{\omega}$ on $\mathcal{B}(H_u)$ satisfying $\bar{\omega}\circ\pi_u=\omega$. Actually $\bar{\omega}$ is a vector state, $\bar{\omega}(T)=\langle T\eta_{\omega},\eta_{\omega}\rangle$, where $\eta_{\omega}\in H_u$. We set $\sigma_*(\omega)=\bar{\omega}$ and extend linearly and this is proven to be a linear isometry that is onto. The adjoint map $\sigma:\pi_u(A)''\to A^{**}$ is what we are looking for.
After proving this, we are able to define multiplication and involution on $A^{**}$ by the preimages through $\sigma$, making it a $C^*$-algebra (a von Neumann algebra actually, since it is $*$-isomorphic to $\pi_u(A)''$. Moreover, the ultraweak topology of $A^{**}$ is simply the weak-* topology it inherits as the dual of $A^*$. Finally, I was able to prove something extra-- that an element $\chi\in A^{**}$ is positive in the $C^*$-algebra sense if and only if $\chi(\tau)\geq0$ for all the positive linear functionals $\tau$ on $A$.
Now I was able to find a bijective linear map $\rho:M_n(A^{**})\to M_n(A)^{**}$. Let $\epsilon_{i,j}:A\to M_n(A)$ denote the positive maps $a\mapsto a\otimes e_{i,j}$ (where $e_{i,j}$ are matrix units). Then we define $$\rho[x_{i,j}]:M_n(A)^*\to\mathbb{C}$$ by $$\rho[x_{i,j}](\Phi)=\sum_{i,j=1}^nx_{i,j}(\Phi\circ\epsilon_{i,j}).$$
The problem is I cannot prove that $\rho$ and $\rho^{-1}$ respect positivity. I guess that the easy half is proving that $\rho$ is positive, but I can't even do that. Here's what I've tried:
It suffices to show that elements of the form $[x_i^*x_j]_{i,j}\in M_n(A^{**})$ are mapped to positive elements of $M_n(A)^{**}$, so it suffices to show that $\rho[x_i^*x_j](\tau)\geq0$ for all states $\tau\in S(M_n(A))$. Computing, we have to show that $\sum_{i,j}x_i^*x_j(\tau_{i,j})\geq0$. Passing everything through the isomorphism $\sigma^{-1}:A^{**}\to\pi_u(A)''$ of Sherman-Takeda, this is equivalent to proving that $$\sum_{i,j}\|\tau\circ\epsilon_{i,j}\|\cdot\overline{\bigg(\frac{\tau\circ\epsilon_{i,j}}{\|\tau\circ\epsilon_{i,j}\|}\bigg)}(B_i^*B_j)\geq0$$
for all operators $B_1,\dots, B_n\in\mathcal{H}_u$, i.e. $$\sum_{i,j}\|\tau\circ\epsilon_{i,j}\|\big\langle B_i^*B_j\eta_{i,j},\eta_{i,j}\rangle_{H_u}\geq0$$ where $\eta_{i,j}$ is the vector that corresponds to the state $\tau\circ\epsilon_{i,j}/\|\tau\circ\epsilon_{i,j}\|$. But unfortunately I can't prove this. As for checking that $\rho^{-1}$ preserves positivity, I have absolutely no clue on how to manage that. Why is this so hard? It feels like this approach should be abandoned from scratch. Does anyone have any idea or a reference to a proof of this result?
Edit: I believe that, using the fact $a\in A_+$ if and only if $\tau(a)\geq0$ for all $\tau\in S(A)$, the problem pretty much reduces to the relations between $S(M_n(A)), S(M_n(A)^{**})$ and $S(M_n(A^{**}))$.
