For the sake of completeness, here are more general statements:

**1) Every separable type I factor is singly generated.** You take the weighted shift $S=\sum_j 2^{-j}E_{j,j+1}$. Then $W^*(S)$ contains $(S^*S)^{1/2}=\sum_j2^{-j}E_{jj}$ and now by using the characteristic function of $\{2^{-j}\}$ we obtain $E_{jj}\in W^*(S)$ for all $j$. Now $E_{k,k+1}=2^kE_{kk}S\in W^*(S)$ and we can get the rest of the matrix units as in $E_{k,k+2}=E_{k,k+1}E_{k+1,k+2}$, etc. Then $W^*(S)$ contains all matrix units and is then equal to $B(H)$.

**2) A countable direct sum of singly generated is singly generated.** If $\mathcal M=\bigoplus_{j\in J}\mathcal M_j$, $J\subset\mathbb N$, we take generators $M_j\in\mathcal M_j$ with $\sigma(M_j)\subset[3/4,1]$ (note that $M$ is a generator if and only if $\alpha M+\beta I$ is a generator for any $\alpha,\beta\in\mathbb C$). Now $M=\bigoplus_j2^{-j}M_j$ is a generator, as we can isolate each summand via functional calculus as in 1).

So 1) and 2) address Sébastien's question.

To go further, a generalization of 1) shows that any separable infinite factor (i.e. II$_\infty$ or III) is singly generated: for type II$_\infty$, you take a dense countable subset of the positive unit ball, normalize all the elements, and form the "weighted shift" as in 1). For type III, you use that $M=M\otimes B(H)$ and you can also form the weighted shift.

**3) Separable abelian von Neumann algebras are also singly generated** (by a selfadjoint), by a Theorem of von Neumann himself. A nice proof can be found in II.2.8 of Davidson's "C$^*$-algebras by Example".

**4) Tensor product of separable abelian and singly generated is singly generated.** This is a wonderful old trick. If $\mathcal A$ is separable abelian, it has a selfadjoint generator $a$. If $\mathcal M$ is singly generated, it has a generator $b+ic$ with $b,c$ selfadjoint. Then $W^*(a)\otimes W^*(b)$ is separable and abelian, and so it has a selfadjoint generator $x$; similarly $W^*(a)\otimes W^*(c)$ has a selfadjoint generator $y$. Then $x+iy$ is a generator for $\mathcal A\otimes\mathcal M$ (since $W^*(x+iy)$ contains $a\otimes I$, and $I\otimes(b+ic)$).

**5) A separable von Neumann algebra with no type II$_1$ summand is singly generated.** Such an algebra is of the form $\bigoplus_j\mathcal A_j\otimes\mathcal M_j$, with each $\mathcal A_j$ abelian and separable, and each $\mathcal M_j$ a non II$_1$-factor (and so, singly generated).

So the "generator problem" for von Neumann algebras is reduced to the case of II$_1$-factors.

**6) The question of whether all separable II$_1$-factors are singly generated is still open.**