Example 12.3.12.label Let $S\suf\cali{B}(\cali{H})$ and consider $S'\define \curl{x\in\cali{B}(\cali{H}):xy=yx,\fall y\in S}$ which is WOT-closed, so if $S=S^{*}$ then $S'$ is a von Neumann algebra. Observe

1. (1)

   $S\suf S''$

2. (2)

   If $S_{1}\suf S_{2}$ then $S_{2}'\suf S_{1}'$

3. (3)

   $S'=S'''$

That is to say the bigger the underlying set $S$ is, the harder it is to “commute” that is belongs to $S'$.

Let $(X,\mu)$ be a probability space, $\cali{H}\define L^{2}(X,\mu)$, and $A=L^{\infty}(X,\mu)$. Consider the map $M:A\to \cali{B}(\cali{H})$ defined by $f\mto M_{f}=(\xi\mto f\cd \xi)$ then

1. (1)

   $\norm{f}{\infty}=\norm{M_f}{}$, in particular $A\suf\cali{B}(\cali{H})$

2. (2)

   $A=A'$ so $A$ is a von Neumann algebra and is a maximal abelian sub-algebra

3. (3)

   $M_{f}^{*}=M_{\cl{f}}$