Positive elements in star algebras

Question

Let $A$ be a $C^\ast$-algebra. Is it possible to prove that if $a \ge 0$ then

$ab, ba \ge 0$ if and only if $b \ge 0$?

Answer 1

No, it is not possible to prove that this is the case.

(non-zero) Counterexample in $\Bbb C^{2 \times 2}$: $$ a = \pmatrix{1&0\\0&0}, \quad b = \pmatrix{1&0\\0&-1} $$

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Counterexample to the comment: $$ a = \pmatrix{1&0\\0&0}, \quad b = \pmatrix{1&1\\1&1} $$ note that neither $ab$ nor $ba$ is self adjoint. Note that $ab$ will be self-adjoint iff $a$ and $b$ commute.

Answer 2

Simplest counterexample: $a=0$.

Somewhat less trivial: in $C([0,1])$ let $a$ be a function that is $0$ on $[0,1/2]$ and nonnegative. Then $ab = ba \ge 0$ if $b \ge 0$ on $[1/2, 1]$, but you could have $b(t) < 0$ somewhere in $[0,1/2)$.