SCHWEITZER 2013 Problem 6

Let ${\mathcal A}$ be a ${C^{\ast}}$ algebra with a unit element and let ${\mathcal A_+}$ be the cone of the positive elements of ${\mathcal A}$ (this is the set of such self adjoint elements in ${\mathcal A}$ whose spectrum is in ${[0,\infty)}$ . Consider the operation $$x \circ y =\sqrt{x}y\sqrt{x},\ x,y \in \mathcal A_+$$ Prove that if for all ${x,y \in \mathcal A_+}$ we have $$(x\circ y)\circ y = x \circ (y \circ y),$$ then ${\mathcal A}$ is commutative.

Proposed by Lajos Molnár