Assume that $A$ is  a  $C^{*}$ algebra with self adjoint elements $A_{sa}$. Assume that for all  $a,b\in A$  we  have $$ab\in A_{sa} \iff  ba \in A_{sa}$$

>Is  $A$ necessarily a  commutative  algebra?

This question is in line of [this post](http://math.stackexchange.com/questions/1687407/is-this-a-characterization-of-commutative-c-algebras)