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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: $x^*x\leq1\to xx^*\leq1$.

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  • $\begingroup$ Surely there must be more hypotheses. For example, if $F$ is a subfield of the complexes closed under complex conjugation, then $F$ and all its matrix rings are Baer*. But a positive answer to Q1 for even a single one of these matrix rings forces $F$ to contain square roots of all of its positive elements. So take $F $ to be the rationals, or any algebraic extension thereof (with a complex embedding). $\endgroup$
    – David Handelman
    Commented Nov 15, 2016 at 0:27
  • $\begingroup$ Instead of algebraic, I meant finite-dimensional. $\endgroup$
    – David Handelman
    Commented Nov 15, 2016 at 0:41

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