Let $n\in\mathbb{N}$ with $n\geq 2$. Let $X_{k}$ be (non-empty) compact Hausdorff spaces for $1\leq k\leq n$. Let $X:=\prod_{k}X_{k}$ and let $\pi_{k}:X\longrightarrow$ be the projections.
It is a straightforward matter to prove that there is a unique linear map
$$\begin{array}[c]{rcl} p &: &\bigotimes_{k}C(X_{k})^{\prime} &\longrightarrow C(X)^{\prime} \end{array}$$
satisfying by $\langle \prod_{k}f_{k}\circ\pi_{k}, p(\otimes_{k}\varphi_{k})\rangle=\prod_{k}\langle f_{k},\varphi_{k}\rangle$ for all $(\varphi_{k})_{k}\in\prod_{k}C(X_{k})^{\prime}$ and $(f_{k})_{k}\in\prod_{k}C(X_{k})$. Moreover it holds that $\|p(\Phi)\|\leq\|\Phi\|_{\pi}$ for all $\Phi\in\bigotimes_{k}C(X_{k})^{\prime}$, where $\|\cdot\|_{\pi}$ is the projective cross norm. Thus $p$ extends uniquely to a continuous linear function
$$\begin{array}[c]{rcl} p &: &\widehat{\bigotimes}_{\pi;k}C(X_{k})^{\prime} &\longrightarrow C(X)^{\prime}. \end{array}$$
I have two questions.
Question 1. Is $p$ in fact an isometric map? It is easy to see that $\|p(\otimes_{k}\varphi_{k})\|=\|\otimes_{k}\varphi_{k}\|_{\pi}$, but I have not yet been able to show that this holds for all $\Phi\in\bigotimes_{k}C(X_{k})^{\prime}$.
(Main) Question 2. Is $p$ surjective? (By the open mapping theorem this would imply that it is open.)
UPDATE According to Example 4.2 and the remarks under Corollary 4.13, as well as Exercise 4.3 of Introduction to Tensor Products of Banach Spaces (Ryan, 2002) the map $p$ is isometric. According to Theorem 4.21 ... if we replace $C(X_{k})$ by reflexive Banach spaces, $E_{k}$, then the map would be surjective if and only if $\widehat{\bigotimes}_{\pi;k}E_{k}$ is reflexive. Since $C(X_{k})$ is not reflexive, I'm not sure what can be applied here.