Suppose that $A$ is a unital $C^*$-algebra, $\varphi\colon A\to B(H)$ is a unital, completely positive map and that $I$ is a non-empty set. If $A\subseteq B(K)$ for some Hilbert space $K$, we can consider the space $A'$ of bounded operators acting on $K\otimes\ell^2(I)$ such that each entry of its matrix representation is an element of $A$.
If we define $\tilde{\varphi}(x)$ for each $x\in A'$ to be the bounded operator on $H\otimes\ell^2(I)$ such that $$\tilde{\varphi}(x)_{i,j}=\varphi(x_{i,j}),$$ we clearly get a unital linear map $\tilde{\varphi}\colon A'\to B(H\otimes\ell^2(I))$. However, I find it hard to prove that this map is also completely positive. It occurred to me that I could maybe use the Stinespring dilation theorem, i.e., that there exists a Hilbert space $H'$, an isometry $V\colon H\to H'$ and a representation $\pi\colon A\to B(H)$ such that $\varphi(x)=V^*\pi(x)V$ for all $x\in A$. However, I run into some problems, namely that I need $\pi$ to be strongly continuous. If $x=a^*a\in M_n(A')$ is positive and we write $a=[a_{kl}]_{k,l=1}^n$, where $a_{kl}=[(a_{kl})_{i,j}]_{i,j\in I}$ for each $k,l=1,\ldots,n$, I then find that $$(x_{k,l})_{i,j}=\sum_{p=1}^n\sum_{q\in I}((a_{pk})_{q,i})^*(a_{pl})_{q,j}\in A,$$ where the series converges in the strong operator topology.
There must be some easier way, but I haven't been able to find one yet...