The center of a von Neumann algebra may or may not have minimal projections

Question

Let $\mathcal H$ be a Hilbert space with $\dim \mathcal H=\infty$. Let $M \subseteq B(\mathcal H)$ be a von Neumann algebra acting on $\mathcal H$. Let $Z$ denotes the Center of $M$. Now a projection $p \in M$ is called Minimal if there exists no non-zero sub-projection of $p$ in $M$, that is, if $0 \ne q$ is a projection in $M$ with $q\le p$ the $q=p$. Now I am thinking that, does $Z$ always have a minimal projection?
If $\dim \mathcal H <\infty$, then what will happen? In case of $B(\mathcal H)$, the rank one projections are minimal projection and hence abelian, but not central since $B(\mathcal H)$ is a factor. But what about an arbitrary von Neumann algebra $M$ on $\mathcal H$. I know when $\dim \mathcal H <\infty$, then $B(\mathcal H)$ can be see as $\mathcal M_n(\mathbb C)$, where $n=\dim(\mathcal H)$. But I have no idea how the von Neumann algebra looks like in case of finite dimension also.
Please help me to solve the above for the both cases where $\dim \mathcal H=\infty$ and $\dim \mathcal H <\infty$. Thank you for your help and time.

Answer 1

As mentioned in the comments, a finite-dimensional von Neumann algebra always has minimal projections; the centre of a finite-dimensional von Neumann algebra is a finite-dimensional von Neumann algebra, which answers the question in that case.

In general, abelian von Neumann algebras need not have minimal projections. Easy example is $L^\infty[0,1]$.