In the case of bounded operators on a Hilbert space $\mathcal{H}$, $L(\mathcal{H})$, there are multiple descriptions of the $\sigma$-strong-* topology, namely:

1) As given by seminorms $p_{\phi},~p_{\phi}^{*}$, indexed by normal linear functionals on $L(\mathcal{H})$, and defined on operators $T\in L(\mathcal{H})$ by $p_{\phi}(T)=\phi(T^{*}T)^{1/2}$ and $p_{\phi}^{*}(T)=\phi(TT^{*})^{1/2}$.

2) As given by semi-norms $p_{\xi_{i}}(T),~p_{\xi_{i}}^{*}(T)$, indexed by sequences $\lbrace \xi_{i}\rbrace_{i\in\mathbb{N}}$ such that $\sum_{i} ||\xi_{i}||\leq \infty$, and defined on operators $T\in L(\mathcal{H})$ by $p_{\xi_{i}}(T)= \sum_{i\in\mathbb{N}}||T\xi_{i}||$ and $p_{\xi_{i}}^{*}(T)=p_{\xi_{i}}(T^{*})$.

3) The strict topology coming from the strict topology when $L(\mathcal{H})$ is identified with the multiplier algebra of the compact operators $C(\mathcal{H})$. More precissely given by the seminorms $p_{K},~p_{K}^{*}$, indexed by operators $K\in C(\mathcal{H})$, and defined on operators $T\in L(\mathcal{H})$ by $p_{K}(T)=||KT||$ and $p_{K}^{*}(T)=||TK||$.

The first description is the usual definition of the $\sigma$-strong-* topology on any W*-algebra, the second is the usual definition when the W*-algebra is faithfully represented on a Hilbert space.

The question I now like to ask, is: "Is there for any W*-algebra $A$ an ideal $I$ such that $M(I)=A$ and the corresponding strict topology coincides with the $\sigma$-strong-* topology?"

May be the standard form of a Von Neumann algebra can be of help, but I am not very into Tomita-Takesaki theory and I am hoping to avoid that.