Let $A$ be a unital Baer \*-ring.


1- Assume that $\{p\_i\}$ is a family of projections in $A$. Let $x$ be an isometry in A (I mean $x^\*x=1$ where $1$ is the unit of $A$). True or false: $\inf (xp\_ix^\*)=x(\inf p\_i)x^\*$ !


2- Let $y$ be an element of $A$. Let us denote $[y]$ by the smallest projection with $[y]y=y$. Let $q$ be a projection in $A$ and assume that $qy=0$. Can we conclude that $q[y]=0$?!



Two points concerning Baer *-rings
A useful reference is Berberian's book "Baer \*-rings".


1 - If $x^\*x=1$, then $x$ is certainly a partial isometry, since $xx^\*x=x$. By taking $e=1$ and $f=xx^\*$ in Proposition 1.9 of Berberian, we obtain a \*-isomorphism $\varphi:A\to fAf$, $a\mapsto xax^\*$ that restricts to an order isomorphism between the projections of $A$ and $fAf$. Since order isomorphisms preserve infima, we obtain $\varphi(\inf\_i p\_i)=\inf\_i \varphi(p\_i)$, i.e., $x(\inf\_i p\_i)x^\*=\inf\_i(xp\_ix^\*)$.


2 - Yes, see Proposition 3.3 of Berberian.


