Two projections P and Q on a Hilbert space \(\mathcal {H}\) are called acute if \(\Vert PQ\Vert <1\) . We utilize the von Neumann alternating projection theorem to prove that if P and Q are acute, then \(P \wedge Q=0\) . Conversely, if \(P\wedge Q=0\) and PQ is a compact operator, then P and Q are acute. An example is presented to show that the assumption of compactness is necessary. Let \(\mathcal {M}\) be a von Neumann algebra, \(\mathcal {M}^{\text {pr}}\) be the lattice of all projections in \(\mathcal {M}\) , and \(P, Q \in \mathcal {M}^{\text {pr}}\) . A pair (P, Q) is called modular in \(\mathcal {M}^{\text {pr}}\) if \((R \vee P)\wedge Q=(R \wedge Q)\vee (P\wedge Q)\) for every \(R\in \mathcal {M}^{\text {pr}}\) with \(R\le Q\) . We present several characterizations of modular pairs of projections in a von Neumann algebra. In particular, for a factor \(\mathcal {M}\) of type I or III, we investigate certain modularity conditions.
