Let $A$ be a Banach algebra with the following property:

*For every two nets $ x_{\alpha}$ and $y_{\alpha}$ in $A$, $x_{\alpha}y_{\alpha}$ converges if and only if $y_{\alpha}x_{\alpha}$ converges.*

Is $A$ necessarily commutative?

1)Note that we do not assume that the above two nets converge to the same value.

2)Note that we do not assume that $A$ has an approximate identity.