Tensor Products of Subspace Lattices and Rank One Density

Abstract

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, and L is a commutative subspace lattice, then L ⊗ M possesses property (p) introduced in [14]. If M is moreover an atomic Boolean subspace lattice while L is any subspace lattice, we provide a concrete lattice theoretic description of L ⊗ 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.