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'?



Action of a group $G$ induces a coaction on $C_0(G)$
The key ideas here are to exploit compactness, and to use a [Partition of unity](https://en.wikipedia.org/wiki/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).




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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](https://doi.org/10.1016/j.jmaa.2010.06.045), also [arXiv:1001.0520](https://arxiv.org/abs/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.


