Von Neumann algebras are $C^*$-algebras

Question

A von Neumann algebra is defined as a *-algebra which is closed w.r.t. the strong operator topology and a $C^*$-algebra as *-algebra with the property $||A||^2=||A^*A||$.

It was said that every von Neumann algebra is a $C^*$-algebra. I tried to show that every von Neumann algebra fulfils the $C^*$ property. I can not get any further and don't know how to use that the von Neumann algebra is closed. Do you have tips?

Thanks for your help.

Answer 1

A von Neumann algebra $M$ lives in some $B(H)$. The C$^*$ relation $$\|T\|^2=\|T^*T\|$$ holds in $B(H)$. So all you need to check is that $M$ is norm closed; and this comes for free since $M$ is closed in a weaker topology than the norm topology.