For a v-N algebra $M$ acting as bounded operators on a separable Hilbert space $H$, how to see $M \rtimes \mathbb{R}$ as a subalgebra of $M \otimes B(L^2(\mathbb{R})$?

Why I am confused is because $x \in M$ is already naturally sitting inside $M \otimes B(L^2(\mathbb{R}))$ as $x \otimes id$, where as it is sitting inside $M \rtimes \mathbb{R}$ as a different object.

In fact action of $x$ on $H \otimes L^2(\mathbb{R})$ is given by the following equation:

$xf(s) = \alpha_{-s}(x) f(s)$ for all $s \in \mathbb{R}, f \in H \otimes L^2(\mathbb{R})$

where $\alpha$ is the action of $\mathbb{R}$ on $M$.