# Existence of (generalized) Crossed Products

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?