Let $H$ and $K$ be two Hilbert spaces. Let us consider two pairs of isometries $(x_1,x_2)$ in $B(H)$ and $(y_1,y_2)$ in $B(K)$ satisfying in $$x_1x_1^*+x_2x_2^*=1_H~~~,~~~y_1y_1^*+y_2y_2^*=1_K$$
Cuntz proved that the C*-algebra generated by $x_1,x_2$ is the same as the C*-algebra generated by $y_1,y_2$.
Q. Is the von Neumann algebra generated by $x_1,x_2$ the same as the von Neumann algebra generated by $y_1,y_2$?
No, they don't have to be isomorphic. Let $\mathcal O_2\subset B(H)$ be an irreducible representation. Then $$ \mathcal O_2''=B(H). $$ Now let $A\subset B(H)$ be a unital, separable, nuclear C$^*$-algebra such that $A''$ is a II$_1$-factor $M$. For instance, we can take $A=UHF(2^\infty)$. Then Kirchberg-Phillips implies that $\mathcal O_2\simeq A\otimes \mathcal O_2$. This means that via this isomorphism we can find isometries $y_1,y_2\in A\otimes\mathcal O_2$ such that $y_1y_1^*+y_2y_2^*=I$, and $C^*(y_1,y_2)=A\otimes\mathcal O_2$. If we consider the irreducible representation $\mathcal O_2\subset B(H)$ as above, we have $$ \{y_1,y_2\}''=C^*(y_1,y_2)''=(A\otimes \mathcal O_2)''= A''\bar\otimes\mathcal O_2''=M\otimes B(H). $$