Let $A$ be a $*$-ring. Let us have some points:

i) We recall that a projection $p$ is a self-adjoint idempotent that is $p=p^*=p^2$.

ii) On the set of projections, we write $p\leq q$ if $pq=p$.

iii) A projection $p$ is called strongly finite if there exist at most finitely many projections $q_1,\cdots,q_n$ with $q_j\leq p$.

iv) For a given $x\in A$, we put $l(x)$ to be the smallest projection with $l(x)x=x$.

Q. Assume that $e$ is an strongly finite projection. Let $p$ be a projection. Is $l(ep)$ an strongly finite projection?

Remark. In this discussion, we assumed $l(x)$ exists for every element $x\in A$. For example $A$ may be assumed a Baer *-ring.


Yes. $l(ep)\leq e$, because $e(ep)=ep$, so if $q\leq l(ep)$, then $$qe=ql(ep)e=ql(ep)=q$$ whence, $q\leq e$, but $e$ is strongly finite, so the number of such $q$'s is finite.


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