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.



characterization of commutative Banach algebras
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.


