TITLE: characterization of commutative Banach algebras
QUESTION [8 upvotes]: 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.

REPLY [10 votes]: No. Some algebras satisfy the identity $xy=-yx$ but are not commutative.  Such an algebra clearly satisfies your condition. Obviously, it never has an approximate identity.
To construct one, let $V,W$ be two Banach spaces and let $f: V \times V \to W$ be a continuous symplectic bilinear form. Then define a multiplication on $V + W$ where anything in $W$ times anything is zero, and the multiplication on $V$ is given by $f$. Then this is anticommutative, and not commutative unless $f=0$. So it provides a counterexample.