For two non-degenerate representations $\pi_j:A\to B(H_{\pi_j})$ ($j=1,2$), we write $\pi_1\sim\pi_2$ if there exists a $w^*$-continuous isometrically isomorphism from $\pi_1(A)''$ onto $\pi_2(A)''$ (in other words, they are the same as von Neumann algebras).
Question: Is this statement true? $\pi_1\sim\pi_2$ if and only if $\ker\pi_1=\ker\pi_2$.
Neither implication is true.
Let $A=\mathbb C\oplus\mathbb C$, and $\pi_j:A\to \mathbb C$ given by $$ \pi_1(a,b)=a,\ \ \ \pi_2(a,b)=b. $$ Then $\pi_1(A)''=\pi_2(A)''=\pi_1(A)=\pi_2(A)=\mathbb C$, but $$ \ker\pi_1=\mathbb C\oplus 0,\ \ \ker\pi_2=0\oplus \mathbb C. $$
The converse is also false. There exist simple C$^*$-algebras (so $\ker\pi_1=\ker\pi_2=0$) with representations that generate non-isomorphic factors. An easy example is $A=UHF (2^\infty) $, where we can have $\pi_1 (A)''$ the hyperfinite II$ _ 1 $-factor and $\pi_2 (A)''$ a type III factor.