A new class of Kadison-Singer algebras

We show that the projection lattice generated by a maximal nest and a rank one projection in a separable infinite-dimensional Hilbert space is in general reflexive. Moreover we show that the corresponding reflexive algebra has a maximal triangular property, equivalently, it is a Kadison-Singer algebra. Similar results are also obtained for the lattice generated by a finite nest and a projection in a finite factor.