Let $M_1$ and $M_2$ be two W*-algebras.
Let $A$ be a C*-algebra and $\pi_j:A\to M_j$ be two faithful representations
with $M_j=\overline{\pi_j(A)}^{w^*}$. Assume that
$$\textrm{The unit of}~ M_j=\textrm{supremum of minimal projections in }~ M_j.$$
Is this statement correct? "W*-algebras $M_1$ and $M_2$ are isomorphic."
For infinite-dimensional separable $H$, take $A$ to be any separable C$^*$-algebra $A\subset B(H)$ with a faithful irreducible representation. For instance $A=K(H)$ or $A=O_2$, the Cuntz algebra.
Let $\pi_1$ be the identity, and $\pi_2$ be the atomic representation (i.e., the direct sum of the GNS representations corresponding to the pure states).
Then $$ \pi_1(A)''=B(H), $$ while $\pi_2(A)''$ is not isomorphic to $B(H)$ since it has central projections.