Let $A$ be a unital Baer*-ring. We say that $a$ is a contraction if $aa^*\leq1$ and $a^*a\leq1$.

Q1) Assume $a$ is a contraction. Has the positive element $1-aa^*$ any square root?
(if yes, seems $1-aa^*$ has as well.)

Q2) Let $x$ be in $A$. True or false: $x^*x\leq1\to xx^*\leq1$.