Proof. (1)$\implies$(2)$\implies$(3) is trivial. Suppose (3) hold then note $A\suf A'$ so let $x\in A''$. We want to show $\fall n\geq 1,\fall \curl{\xi_i}_{1}^{n}\suf\cali{H},\fall \e>0,\exists y\in A:\norm{(x-y)\xi_i}{}<\e,\fall i\leq n$. First assume $n=1$, fix $\xi\in\cali{H}$ and let $K=\cl{A\xi}$. Let $p=proj_{K}\in\cali{B}(\cali{H})$ then

1. (1)

   $K$ is $A$-invariant

2. (2)

   $K^{*}$ is $A$-invariant

then $p\in A'$ so $px=xp$. Now $\xi=1\xi\in A\xi\in K$ so $p\xi=\xi$ so $p(x\xi)=xp\xi=x\xi$ showing $x\xi\in K$. In general, fix $\xi_{1},...,\xi_{n}\in\cali{H}$ and consider $\cali{H}^{\op n}$ and let $\tea:\cali{B}(\cali{H})\to \cali{B}(\cali{H}^{\op n})$ be the natural map. Set $\cali{B}=\tea(A)$ then $\tea(x)\in \cali{B}''$.$\square$