Let $0\le y \le x \in B(H)$. It is known that there exists an operator $a$ from the unit ball of $B(H)$ such that $y=axa^*$. I am thinking on the supports of $a$. For example, if the supports of $y$ and $x$ are both $1$, then can we say that the (left and right) supports of $a$ are $1$?
It seems that if the following question has an affirmative answer, then the above question also has an affirmative answer.
Let $x$ be a bounded linear operator on a Hilbert space $H$. We denote by $r(x)$ the right support projection of $x$. Let $\{p_n\}$ be a sequence of projections increasing to the identity $1$. Assume that the right support $r(xp_n)= p_n$ for every $n$. Is it correct that $r(x)=1$?