Abelian von Neumann algebra

Question

Let $M$ be a family of commutative normal operators which is closed under adjoint. clearly $M\subset M'$, but I do not know why $M'\subset M^{''}$? and how can conclude that $M^{''}$ is abelian?

Answer 1

It is not true that $M'\subset M''$, for example consider $M=\{I\}$ the identity operator. To show that $M''$ is commutative though just use the fact that it is the WOT closure of the algebra generated by $M$ (the Double Commutant theorem.)