Tensor products of subspace lattices and rank one density

Abstract

We show that, if $M$ is a subspace lattice with the property that the rank one subspace of its operator algebra is weak* dense, $L$ is a commutative subspace lattice and $P$ is the lattice of all projections on a separable infinite dimensional Hilbert space, then the lattice $L\otimes M\otimes P$ is reflexive. If $M$ is moreover an atomic Boolean subspace lattice while $L$ is any subspace lattice, we provide a concrete lattice theoretic description of $L\otimes M$ in terms of projection valued functions defined on the set of atoms of $M$. As a consequence, we show that the Lattice Tensor Product Formula holds for $\Alg M$ and any other reflexive operator algebra and give several further corollaries of these results.