Let $A \subset B(H)$ be a unital $C^\ast$-algebra and $\theta: G \rightarrow \mathrm{Aut}(A)$ an action and let $\omega: G \times G \rightarrow U(\mathcal{Z}(A))$ be a $2$-cocyle with respect to $\theta$ taking values in the unitary group of the center of $A$, i.e. a function satisfying $$ \omega(g,h) \, \omega(g h, k) = \theta_g(\omega(h,k)) \, \omega(g, hk). $$ We can define the twisted (reduced) crossed product $A \rtimes_{(\theta, \omega)} G \subset B(L_2(G;H))$ as the smallest $C^\ast$-algebra generated by $A$, embedded inside $B(L_2(G;H))$ like in the case of crossed products, and the unitaries $\{u_g\}_{g \in G}$ which act by $$ (u_g \xi)(h) = \omega(h^{-1}, g) \, \xi(g^{-1} h), $$ for $\xi \in L_2(G;H)$. Similarly we can define a full/universal crossed product $A \rtimes_{(\theta, \omega)}^{ full} G$ by taking the supremum over all representation preserving the multiplication law $$ (x \, u_g) \, (y \, u_h) = x \, \theta_g(y) \, \omega(g,h) \, u_{g \, h}. $$ What can be said on the amenability of $A \rtimes_{\theta, \omega} G$? In particular.
Question 1.: Is there any definition of amenability for $(\theta,\omega)$ in the literature satisfying that, if $(\theta,\omega)$ is amenable, then
$A \rtimes_{(\theta, \omega)} G$ is nuclear whenever $A$ is.
$A \rtimes_{(\theta, \omega)} G = A \rtimes_{(\theta, \omega)}^{full} G$
Question 2.: Is that notion of amenability equal to the amenability of $\theta$? (in the sense of [1, Definition 4.3.1]) Or, by the contrary, can an amenable action have nonnuclear twisted crossed products?
[1] Brown, Ozawa. $C^\ast$-Algebras and Finite-Dimensional Approximations.
