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Assumptions
Let \(\pi:X\twoheadrightarrow Y\) be an equicontinuous extension of compact Hausdorff $G$-spaces and let \(C(X), C(Y)\) be commutative and unital (or just commutative and nuclear?).
We assume that the map \begin{equation} \label{eqn:assumed} (ℓ^∞(G) ⊗ C(Y)) \rtimesmaxG → ℓ^∞(G,C(Y)**)\rtimesmaxG \end{equation} is injective.
The following canonical map is injective: \[ (\ell^\infty(G) \otimes C(X)) \rtimes_{\max} G \to (\ell^\infty(G)\otimes \ell^\infty(Y;X))\rtimes_{\max} G \]
This is the proof