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A commutative algebra

Source: Miklós Schweitzer 2013, P6

July 12, 2014
geometry3D geometryadvanced fieldsadvanced fields unsolved

Problem Statement

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