Tensor products of subspace lattices and rank one density

TL;DR

This paper investigates the reflexivity of tensor products of subspace lattices, establishing conditions under which the lattice tensor product formula holds, with new descriptions for atomic Boolean cases and implications for operator algebra theory.

Contribution

It provides new conditions for reflexivity of tensor products of subspace lattices and a concrete lattice description for atomic Boolean cases, extending the lattice tensor product formula.

Findings

Reflexivity of tensor products under certain conditions.

Concrete lattice description for atomic Boolean subspace lattices.

Validation of the lattice tensor product formula for reflexive operator algebras.

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.