The following lemma appears in the paper "Rokhlin dimension for flows"- by Winter, Hirshberg, Szábo, Wu.
Lemma 6.5: Let $A $ be a $\sigma $-unital $C^*$-algebra with a flow $\alpha:\Bbb {R}\to Aut (A) $. Then $A\rtimes_{\alpha} \Bbb {R}$ is $\sigma $-unital and there is a countable approximate unit of $C^*(\Bbb{R}) $ consisting of functions in the convolution subalgebra of $L^1 (\Bbb {R}) $ whose Fourier transform has compact support, $(a_j)_{j\in\Bbb {N}} $ is a countable approximately invariant approximate unit of $A $, and they satisfy $[a_j,g_j] \to 0$ as $j\to 0$.
The proof should be standard. What have I done so far:
It is known that we can always choose an approximately invariant countable approximate unit for $A $.
Let $\mathcal {U} $ be any countable neighborhood basis of $0$ in $\Bbb{R} $ (e.g. open balls of radiuses $1/n $). For each $u\in \mathcal {U} $, let $f_u \in C_c (\Bbb {R})^{+} $ be s.t. $supp{f_u}\subseteq u $ and $\int_{\Bbb {R}}f_u (t)dt=1$. (Here maybe it could be better to try to construct such $f_u$'s with $supp (\hat {f_u})\subseteq u $). Let $\Lambda=\Bbb {N} X \mathcal {U} $ be a directed set with order $(n,u)\geq (m,v) $ if $n\geq m $ and $u\subseteq v $. Then it is easy to verify that $(f_u\otimes a_n)_{(n,u)\in \Lambda}$ is an approximate unit for the crossed product.
However, I have no control on the Fourier transform of these functions and I'm not sure what is the last property should be satisfied (where is $a_jg_j $? Why don't they always commute ?)
Thank you for the help
