Let $A$ be a unital $C^*$ algebra such that for any two positive elements $x$, $y$ in $A$, whenever $x\leq y$ we have that $x^2\leq y^2$. Prove that $A$ is abelian.
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asked Jun 29, 2019 at 8:37
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6$\begingroup$ Is this homework? I don't know the solution off the top of my head, but problems phrased in the imperative often are. $\endgroup$– LSpiceCommented Jun 29, 2019 at 11:30
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$\begingroup$ This is not a homework. I am reading Jesse Petersen's notes on von Neumann algebras. I have done other exercises. But I could not solve this one. $\endgroup$– A beginner mathmaticianCommented Jun 29, 2019 at 12:00
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9$\begingroup$ Right; usually exercises from a book are better received if you (1) indicate that they are exercises and (2) say what you've tried. $\endgroup$– LSpiceCommented Jun 29, 2019 at 12:17
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8$\begingroup$ See Proposition 1.3.9 of "C*-Algebras and Their Automorphism Groups" (second edition) by G. K. Pedersen. It takes a little work. $\endgroup$– Nik WeaverCommented Jun 29, 2019 at 15:54
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