I am a beginner in Operator Theory and Functional Analysis.
On the space of bounded operators on Hilbert space $H$, We claim that " It is true that von Neumann algebras are $C^*$- algebras of operators".
It is obviously true that any von Neumann algebra is close under adjoint operation.
In von Neumann algebras, we use weak operator topology. In $C^*$ - algebra, we use norm topology. In general, weak opeator convergence does not imply norm convergence.
I wonder how can we say that they are equivalent in the space of bounded operators?
Thank you for your time.