Tensor Products of Subspace Lattices and Rank One Density

Abstract

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