I am looking for any information concerning atomic W*-algebras.
Def. A W*-algebra $M$ is called atomic if for any projection $p$ in $M$ there is a minimal projection $e\in M$ with $e\leq p$.
Q. Is there any characterization for atomic W*-algebras?
Let me say what I know about. Let Min$(M)$ be the set of all minimal projections in $M$. For a given minimal projection $e$ in $M$, let $z(e)$ be the central carrier (central support) of $e$ ($z(e)$ is minimal among all central projections). The W*-algebra $z(e)M$ is a factor and $M=\bigoplus_{e\in Min(M)} z(e)M$. To sum up with, any characterization of atomic+factor W*-algebras is concerned. Any reference or direct proof is appreciated.
An atomic factor is necessarily $B(H)$.
Because of the lack of central projections, a factor is of a definite type. Because it has a minimal projection, it has to be type I.