Let $M$ be a finite dimensional von-Neumann algebra. We know this algebra is generated by its projections. My question maybe simple. Can one computing these projections? What about its minimal or central projections?
If possible please give me a reference for this.
Thanks
You would have to define "compute". To "compute" an element of an algebra (or any other mathematical object, for that matter) you need to have some kind of presentation of the object.
In the case of a finite-dimensional von Neumann/C$^*$-algebra, one can prove that they are always (isomorphic to) a direct sum of full matrix algebras. Are you starting with the former or the latter?
And again with the word "compute". Can you compute all rank-2 orthogonal projections in $M_3(\mathbb C)$? It depends on what you mean. They are all of the form $vv^*+ww^*$ with $v,w$ unit mutually orthogonal vectors in $\mathbb C^3$, so in a sense they are very explicit. On the other hand, I'm not saying what $v$ and $w$ are. If you try to somehow parametrize them, it can probably be done, but it is not pretty and the general expression for $vv^*+ww^*$ will be headache that likely contributes nothing.
Anyway, I haven't really answered your question. It depends on how you express your algebra, and what you mean by "compute".