Let $A$ be a Baer *-ring. Let $x$ be an isometry (meaning $x^*x=1$ where $1$ is the unit of $A$).
Let $e$ be a finite projection in $A$ such that $ex^ne=ex^n$ for every $n\geq0$.

Q. Can we say that $e\leq \inf x^nx^{*}$

Remark. $x^nx^{*n}$ is a projection for every $n\geq1$.

Let $A$ be a Baer *-ring. Let $x$ be an isometry (meaning $x^*x=1$ where $1$ is the unit of $A$).
Let $e$ be a finite projection in $A$ such that $ex^ne=ex^n$ for every $n\geq0$.

Q. Can we say that $e\leq \inf x^nx^{*n}$

Remark. $x^nx^{*n}$ is a projection for every $n\geq1$.