Let $M$ be a von Neumann algebra. We say that $M$ has many sufficiently normal irreducible representations, namely $\{\pi_i\}$, if $||a||=\sup ||\pi_i(s)||$. For example the second dual of a C*-algebra does.
I am looking for an equivalent condition for this property in term of type theory or some thing else.
Question: $M=?$ if and only if $M$ has many sufficiently normal irreducible representations.
An irreducible representation $\pi:M\to B(K)$ has range dense in $B(K)$; but, if $\pi$ is normal, the range is all of $B(K)$. As $\ker\pi$ is a weakly closed ideal of $M$, it is of the form $pM$ for some central projection $p\in M$, and we get $(1-p)M\simeq B(K)$.
In other words, $M$ has sufficiently many normal irreducible representations if and only if $M$ is a direct sum $\bigoplus B(K_j)$.