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Edit: closed convex hull added.

I am trying to understand the state space of $C(K,M_n)=C(K)\otimes M_n$ for $K$ a compact space. My guess would be that these are the closed convex hull of states on $C(K)$ (=probability measures) paired with the compression by some continuous function $v:K\to \mathbb{C}^n$ in the sense $$ (v^*fv)(x)= v(x)^*f(x)v(x). $$ So both together would be a composition of the ucp-map $$v^*(\cdot)v:C(K,M_n)\to C(K)$$ followed by some state $\varphi: C(K)\to \mathbb{C}$.

But will all states really lie in the closed convex hull of such resp. does someone know how the state space looks like?

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Strictly speaking it wouldn’t be necessary to ask the vector function itself to be continuous (certainly the applied compression being measurable would suffice) but I guess one could arrange somehow by replacing (unless some topological obstruction lurking behind) the vector function itself to be continuous.

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  • $\begingroup$ Good point, thanks! So I added on their closed convex hull. $\endgroup$
    – C-star-W-star
    Commented May 26, 2022 at 13:35
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    $\begingroup$ The states on $C(K,M_n)$ are exactly the Radon measures on $K$ that take values in positive linear functionals on $M_n$, and are normalized such that the measure of $K$ itself is a state on $M_n$. One way to see this is to simply look at a state entry-wise on the matrix, which is then a bounded linear functional on $C(K)$. $\endgroup$
    – Tobias Fritz
    Commented May 26, 2022 at 13:54
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    $\begingroup$ By the way, are you working on a moment problem on $C^*(\mathbb{Z}/2 \times \mathbb{Z}/2)$? I solved one such problem myself in this paper, so perhaps it's related. (It's been my first paper ever, and the writing is a bit quirky... See in particular App A in there.) $\endgroup$
    – Tobias Fritz
    Commented May 26, 2022 at 13:54
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    $\begingroup$ Hi Tobias, indeed we’re working on such a problem. Also thanks for the suggestion about your paper! :) $\endgroup$
    – C-star-W-star
    Commented May 27, 2022 at 8:04

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