Towards the carpenter's theorem

Let $\mathcal{M}$ be a II$_1$ factor with trace $\tau$, $\mathcal{A}\subseteq \mathcal{M}$ a masa and $E_{\mathcal{A}}$ the unique conditional expectation onto $\mathcal{A}$. Under some technical assumptions on the inclusion $\mathcal{A}\subseteq \mathcal{M}$, which hold true for any semiregular masa of a separable factor, we show that for elements $a$ in certain dense families of the positive part of the unit ball of $\mathcal{A}$, it is possible to find a projection $p\in\mathcal{M}$ such that $E_{\mathcal{A}}(p)=a$. This shows a new family of instances of a conjecture by Kadison, the so-called ``carpenter's theorem''.