Action of a group $G$ induces a coaction on $C_0(G)$ In this question, I follow the book "An invitation to quantum groups and duality" by Timmerman, p259.
Let $G$ be a locally compact group and $C$ be a $C^*$-algebra. Assume an action
$$\alpha: G \to \operatorname{Aut}(C)$$
is given. Define a $*$-homomorphism
$\delta_0: C \to C_b(G,C)$ by $\delta_0(c)(g) = \alpha_g(c)$. I try to understand the proof of theorem 9.2.4. In particular, I want to prove that
$\delta_0(C)C_0(G)$ is dense in $C_0(G,C).$
Note that we can embed $C_0(G)\hookrightarrow C_0(G, M(C))$ via $f \mapsto f(-)1_{M(C)}$ where $M(C)$ denotes the multiplier algebra of $C$. Then $\delta_0(C)C_0(G)$ is a multiplication in $C_0(G,M(C))$ which returns an element of $C_0(G,C)$ since $C$ is an ideal in $M(C)$.
Why is the density claim true? Timmerman writes:

The equation $\delta_0(\alpha_{g^{-1}}(c))(g) = \alpha_g(\alpha_{g^{-1}}(c)) = c$ shows that for every $g \in G$ and every $c \in C$ there is $f \in \delta_0(C)$ with $f(g) = c$. Now the density follows from a standard argument.

What is this 'standard argument'?
 A: The key ideas here are to exploit compactness, and to use a Partition of unity.  In fact, I think the most useful formulation of this, for locally compact spaces, can be found in Rudin, "Real and Complex Analysis", Theorem 2.13:

Let $G$ be a locally compact space, $K\subseteq G$ compact, and let $K\subseteq U_1\cup\cdots\cup U_n$ be a finite open cover.  There are continuous functions $f_i:G\rightarrow [0,1]$ with compact support contained in $U_i$, and with $\sum_i f_i(x) = 1$ for each $x\in K$.

We call the $(f_i)$ a partition of unity subordinate to $(U_i)$.
Let $X$ be the closed linear span $\delta_0(C) C_0(G)$, a $C_0(G)$-submodule of $C_0(G,C)$.  What Timmermann shows is that for each $g\in G,c\in C$ there is some $f = f_{g,c}\in X$ with $f(g)=c$.  Here is a sketch of how to proceed:

*

*Let $K\subseteq G$ be compact and fix $c\in C$.  For each $g\in K$, the function $f_{g,c}$ is continuous.  Thus, for $\epsilon>0$, there is an open cover $(U_g)_{g\in K}$ of $K$ with $h\in U_g \implies \|f_{g,c}(h) - c\|<\epsilon$.

*By compactness, find finitely many points $(g_i)$ with $(U_{g_i})$ a cover of $K$.

*Pick a partition of unity subordinate to $(U_{g_i})$, say $(g_i)$.

*Set $f = \sum g_i f_{g_i,c}$

*For $h\in K$, if $h\in U_{g_i}$ then $\|f_{g_i,c}(h)-c\|<\epsilon$, while if $h\not\in U_{g_i}$ then $g_i(h)=0$.  Conclude
$$ f(h) = \sum_i g_i(h) f_{g_i,c}(h) = \sum_{h\in U_i} g_i(h) f_{g_i,c}(h)
\approx_\epsilon c, $$
using that $\sum_i g_i(h)=1$.

Identify $C_0(G) \odot C$ with a subspace of $C_0(G,C)$.  We have shown that if $g\in C_0(G)$ has compact support and $c\in C$ then $g\otimes c\in X$.  Taking the closure, conclude that $C_0(G)\otimes C\subseteq X$.  So now take your favourite proof that $C_0(G)\otimes C \cong C_0(G,C)$ to conclude that $X = C_0(G,C)$.  (Your "favourite proof" will also use a partition of unity argument).

Actually, I think there is a much nicer, and well-explained, but still elementary argument given by Soltan in "Examples of non-compact quantum group actions", doi.org/10.1016/j.jmaa.2010.06.045, also arXiv:1001.0520.  See Proposition 2.1.  In fact, in Section 2 of this paper is developed some general theory, and Proposition 2.3 gives a somewhat more "conceptual" (and easier) proof of the fact you are after, using some general theory.
