In this post I'm trying to understand a small but significant part of an article by Corran Webster and Soren Winkler regarding a matrix convex generalization of the Krein-Milman theorem. (see the paragraph before Proposition 3.5 in Section 3 of the original article).
Now
``It is now possible to identify $M_r(A(\mathbf{K},\mathbb{C}))$ and $A(\mathbf{K},\mathbb{M}_r)$, and we may thus use the ordering on $A(\mathbf{K},\mathbb{M}_r)$ to define a positive cone in $M_r(A(\mathbf{K},\mathbb{C}))$''
Here, $\mathbf{K}=(K_n)_{n=1}^{\infty}$ is a compact matrix convex set over some locally convex vector space $V$ and $\mathbb{M}_r=(M_n(M_r))_{n=1}^{\infty}$. $A(\mathbf{K},\mathbb{C})$ is the set of matrix affine functions $F=(F_n)_{n=1}^{\infty}$, where $F_n:K_n\to M_n$, and $F_1$ is continuous.
It seemed quite natural at first glance but I am not sure how is the identification made. Given some $(F_{ij})\in M_r(A(\mathbf{K},\mathbb{C}))$, I can try to define some $\tilde{F}\in A(K,\mathbb{M}_r)$ by defining its first level. Namely $\tilde{F}_1(k)=(F_{ij}(k))$ for all $k\in K_1$. But trying to define it for $K_n$ for $n>1$ turns out harder then I thought (and this is something that was left for self checking in the article and still bugs me).
Namely, for $k\in K_n$ I can define $\tilde{F}_n(k)=(F_{ij,k})\in M_n(M_r)$ and use the usual identification to swich to $M_r(M_n)$. The only problem that I'm facing, is knowing that this map with the switch is matrix affine, and it seems that it is not (because each $r\times r$ block in $M_n(M_r)$ is the image of a matrix affine map, but now I'm mixing their elements).
Any help with what I am missing here would be appreciated.
To summarize, I am trying to find and understand a way to identify $M_r(A(\mathbf{K},\mathbb{C}))$ and $A(K,\mathbb{M}_r)$, in order to get a better description of what is the positive cone in the first set.
