Assume that $\mathcal{A}$ is a separably acting von Neumann algebra with direct integral decomposition into factors $\mathcal{A}_{\gamma}$ for $\mu$-a.a. $\gamma\in\Gamma.$
$$\mathcal{A}\cong \int_{\Gamma}^{\oplus} \mathcal{A}_{\gamma}\,d \mu(\gamma).$$
Can we claim that $$\mathcal{A}\bar \otimes \mathcal{B}(\ell^2(\mathbb{N}))\cong \int_{\Gamma}^{\oplus} \mathcal{A}_{\gamma}\bar\otimes \mathcal{B}(\ell^2(\mathbb{N}))\,d \mu(\gamma) \,\,\,\,? $$
I found a possible answer in Effro's paper but still I am not so sure.[Effros-global structure][1]
[1]: https://community.ams.org (page 10, relation (2)).