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



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> Is $A$ necessarily a commutative algebra?
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This question is in line of [this post](https://math.stackexchange.com/questions/1687407/is-this-a-characterization-of-commutative-c-algebras)



Is this a characterization of commutative $C^{*}$ algebras?
Yes. I will show that any two positive elements of $A$ commute. Since every element is a linear combination of positive elements, this suffices.


Say $a$ and $b$ are positive. Then $a^{1/2}ba^{1/2} \in A\_{sa}$, so by hypothesis $ba^{1/2}a^{1/2} = ba \in A\_{sa}$. That is, $ba = (ba)^\* = a^\*b^\* = ab$. QED


