Some new classes of Kadison-Singer lattices in Hilbert spaces

Let $\mathcal{N}$ be a maximal and discrete nest on a separable Hilbert space $\mathcal{H},E_\xi$ the projection from $\mathcal{H}$ onto the subspace ?ξ] spanned by a particular separating vector ξ for $\mathcal{N}'$ and Q the projection from $\mathcal{K} = \mathcal{H} \oplus \mathcal{H}$ onto the closed subspace $\left\{ {\left( {\eta ,\eta } \right):\eta \in \mathcal{H}} \right\}$ . Let $\mathcal{L}$ be the closed lattice in the strong operator topology generated by the projections and Q. We show that $\mathcal{L}$ is a Kadison-Singer lattice with trivial commutant, i.e., $\mathcal{L}' = \mathbb{C}I$ . Furthermore, we similarly construct some Kadison-Singer lattices in the matrix algebras M _2n (?) and M _2n?1(?).