Topologies on the automorphism group of a von Neumann algebra

Question

Assume that $\varphi$ is a faithful normal state on a von Neumann algebra $M$, and define the norm $\|x\|_{2,\varphi}^\# = \varphi(x^\ast x+x x^\ast)^{1/2}$. Consider the following three topologies on Aut$(M$), defined by stating when a net of automorphisms $\Phi_i$ converges to some automorphism $\Phi$.

1. $\Phi_i \rightarrow \Phi$ if and only if $\| \omega \circ \Phi_i - \omega \circ \Phi\| \rightarrow 0$ for all $\omega \in M_\ast$.
2. $\Phi_i \rightarrow \Phi$ if and only if $\Phi_i(x) \rightarrow \Phi(x)$ $^\ast$-strongly for all $x \in M$.
3. $\Phi_i \rightarrow \Phi$ if and only if $\|\Phi_i(x)-\Phi(x)\|_{2,\varphi}^\# \rightarrow 0$ for all $x \in M$.

Are the three topologies equivalent when we view $M \subset \mathcal{B}(L^2(M,\varphi))$?

If not, are these notions of convergence equivalent, if we consider a group homomorphism $G \rightarrow \text{Aut}(M)$ with $G$ a Polish group?