Compare the following construction with the crossed product construction for $C^*$-dynamical systems:

Let $W\curvearrowright X$ be a group action of a discrete group on a compact Hausdorff space $X$ and let $\pi: \mathbb{C}[W] \rightarrow \cal B(H)$ be a cyclic representation of the group algebra $\mathbb{C}[W]$ such that $w\mapsto \pi (w)\xi $ is injective. Further assume that $\nu: C(X) \rightarrow \cal B(K)$ is a faithful representation of the $C^*$-algebra $C(X)$.

Now consider the map

$\overline{\nu}: C(X) \rightarrow \cal B(H\otimes K)$ given by $\overline{\nu}(f) (\pi (w)\xi \otimes \eta) :=\pi(w)\xi \otimes \nu(w^{-1} f) \eta$

where $\xi$ is the cyclic vector and $\eta \in \cal K$. I am wondering under which conditions this map $\overline{\nu}$ (which is a priori not necessarily well-defined) actually exists (i.e. under which conditions is $\overline{\nu}(f)$ for every $f\in C(X)$ a well-defined bounded operator)?

In the case that $\pi$ is the left-regular representation, this is fulfilled. The same is true if the action $W\curvearrowright X$ is trivial. But I do not expect that $\overline{\nu}$ is well-defined for any representation and group action. Can we characterize this property or give cases in which it holds?


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