von Neumann algebra

Question

Let $M$ be a subset of $B(\mathcal{H})$ (the space of bounded linear operators) such that $M'$ is a von Neumann algebra.

As we know if $M$ is invariant under involution, then $M'$ is a von Neumann algebra. My question is about the converse of it. Is $M$ invariant under $*$-operation, if $M'$ is a von Neumann algebra?

Answer 1

Fix some $T\in B(\mathcal H)$ that is normal but not self-adjoint, and put $M=\{T\}$. Then $M'$ is a von Neumann algebra, but $M$ is not self-adjoint.