Exercise 15, § 6

a) Let A be a commutative star algebra. If $x$ and $y$ are positive elements of A such that $x\leqslant y$, then $x^2\leqslant y^2$.

b) There exists a star algebra A and elements $x$ and $y$ of A such that $0\leqslant x\leqslant y$, but such that $x^2$ is not $\leqslant y^2$.

c) There exists a star algebra A and elements $x$ and $y$ in A such that $|x+y|$ is not $\leqslant |x|+|y|$.