3 problems

Let $\mathcal{M}$ be a separable factor. An operator $T\in\mathcal{M}$ is reducible if there is a projection $P\in\mathcal{M}$ with $P\ne 0,I$ and $PT=TP$. Halmos' conjecture. The…

Edge-disjoint 1-factor conjecture. Some realization of $\mathrm{\pi}$ has $k$ edge-disjoint $1$-factors if and only if

Anti-directed 2-factor conjecture. If