# Is the double commutant $A''$ commutative if $A$ is commutative?

If $A$ is a commutative C*-subalgebra of linear bounded operator space $B(H)$ on some Hilbert space $H$, so is the double commutant $A''$. It follows from $A$ is dense in $A''$ and the multiplication is continuous on each factor respectively, respect to the strong operator topology.

But note that "double commutant" and "commutative" are both algebraic terms, I want to ask:

1. Can the assertion "$A''$ is commutative" be verified in an algebraic way?

2. Or, a more general question: If $A$ is a commutative subalgebra of an algebra $B$, is the double commutant $A''$ of $A$ also commutative?

If $A$ is commutative, you have $$A\subset A'.$$ Then $$A'\supset A''.$$ And then $$A''\subset A'''.$$ So $A''$ is contained in its commutant and is thus commutative.
The above reasoning shows that all even higher commutants of $A$ are commutative.