Let $\mathcal{M}$ be a finite dimensional von Neumann algebra, then : 
$$\mathcal{M} \simeq \bigoplus\_i M\_{n\_i}(\mathbb{C})$$



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> **Question** : Is it singly generated (as von Neumann algebra)? how ?
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**Allowed operations** : $() \mapsto I$ , $(A) \mapsto \lambda A$ or $\mathbf{A^\*}$ and $(A,B) \mapsto A+B$ or $A B$ 



Are the finite dimensional von Neumann algebras, singly generated?
Pick distinct complex numbers $\lambda\_1,\ldots,\lambda\_k$ and consider the element
$$
X:=\Bigg(\,\underbrace{\begin{smallmatrix}
\lambda\_1&1&0&0&0\\
0&\lambda\_1&1&0&0\\
0&0&\ddots&\ddots&0\\
0&0&0&\lambda\_1&1\\
0&0&0&0&\lambda\_1\\
\end{smallmatrix}}\_{n\_1}\,\Bigg)
\oplus
\Bigg(\,\underbrace{\begin{smallmatrix}
\lambda\_2&1&0&0&0\\
0&\lambda\_2&1&0&0\\
0&0&\ddots&\ddots&0\\
0&0&0&\lambda\_2&1\\
0&0&0&0&\lambda\_2\\
\end{smallmatrix}}\_{n\_2}\,\Bigg)
\oplus
\cdots\oplus
\Bigg(\,\underbrace{\begin{smallmatrix}
\lambda\_k&1&0&0&0\\
0&\lambda\_k&1&0&0\\
0&0&\ddots&\ddots&0\\
0&0&0&\lambda\_k&1\\
0&0&0&0&\lambda\_k\\
\end{smallmatrix}}\_{n\_k}\,\Bigg)
$$
I claim that for generic $\lambda\_i$ that element generates your algebra.


First of all, $(X-\lambda\_j)^{n\_j}$ has zero for its $j$th component.
Taking products of such elements, one can achieve something whose only non-zero entry is in a given summand (say the $i$th summand). The remaining term is Toeplitz and upper triangular. More precisely, it is given by
$$
A:=\left(\begin{matrix}
a&b&c&d&e\\
0&a&b&c&d\\
0&0&\ddots&\ddots&c\\
0&0&0&a&b\\
0&0&0&0&a\\
\end{matrix}\,\right)
$$
with $a=\prod\_{j\not =i}(\lambda\_i-\lambda\_j)^{n\_j}$
and $b=$ (some horrible expression which is obviously positive if all the $(\lambda\_i-\lambda\_j)$ are positive, and therefore non-zero if the $\lambda\_i$ are chosen generically).


Then it's a matter of playing around to see that the above matrix and its adjoint generate all of $M\_{n\_i}(\mathbb C)$.
The first operation is to consider $N:=A^2-aA$, which is nilpotent with non-zero terms on the subdiagonal. Then you take an appropriate linear combination of powers of $N$ to generate the element
$$
\left(\begin{matrix}
0&1&0&0&0\\
0&0&1&0&0\\
0&0&\ddots&\ddots&0\\
0&0&0&0&1\\
0&0&0&0&0\\
\end{matrix}\,\right).
$$
Then you take the transpose of that element, and you're essentially done.


