Let $A$ be a Baer*-ring with the unit $1$. An important point that holds in every Baer*-ring is this: for every element $y\in A$ there is an smallest projection $e_y\in A$, called the left projection of $y$, satisfying in $e_yy=y$.
We say $x\in A$ is a shift if $x^*x=1$ and there exists a sequence of pairwise orthogonal projections $\{p_n\}$ such that:
$\sup_{n\geq1} \sum_1^n p_k=1$.
$p_n=e_{xp_{n-1}}$.
Q. Let $x$ be a shift. For a finite projection $q$ in $A$, I guess $q\wedge e_{x^*q}$ should be strictly less than $q$, however I have no clear proof!
