Let $A$ be a $C$*-algebra and $p,q\in A^{**}$ be commuting projections. Then there exist self-adjoint nets $(x_i)_i$ and $(y_j)_j$ in $A$ with $x_i\to p$ and $y_j\to q$ in the weak *-topology. Can these nets be chosen such that $x_i y_j=y_j x_i$ for all $i$ and $j$?
Extra question: is this true if $A$ is a $JB$-algebra?
Yes. Let $1$ be the unit of $A^{**}$. Let us put
$$e=p\wedge q~~~,~~~f_1=(1-p\wedge q)\wedge p~~~,~~~f_2=(1-p\wedge q)\wedge q$$
Then $\{e,f_1,f_2\}$ are pairwise orthogonal and $$p=e+f_1~~~,~~~q=e+f_2$$
There are bounded nets $\{a_i\}$, $\{b_i\}$ and $\{c_i\}$ in $A$ with
$$e=w^*-\lim a_i~~,~~f_1=w^*-\lim b_i~~,~~f_2=w^*-\lim c_i$$ such that $a_i\leq e,b_i\leq f_1~,~c_i\leq f_2$(for all $i)$.
The nets $\{x_i=a_i+b_i\}$ and $\{y_i=a_i+c_i\}$ work for this problem.