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”.