6

Part of 2013 Miklós Schweitzer

Problems(1)

A commutative algebra

Source: Miklós Schweitzer 2013, P6

{\mathcal 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

\displaystyle x \circ y =\sqrt{x}y\sqrt{x},\ x,y \in \mathcal A_+

Prove that if for all

{x,y \in \mathcal A_+}

\displaystyle (x\circ y)\circ y = x \circ (y \circ y),

{\mathcal A}

is commutative.
Proposed by Lajos Molnár

geometry3D geometryadvanced fieldsadvanced fields unsolved