The tensor product of a CSL and an ABSL

Abstract

We study the question that asks whether the tensor product of two reflexive subspace lattices is reflexive. In particular, we study the tensor product of a commutative subspace lattice L and an atomic boolean subspace lattice M and we prove that it is equal to the extended tensor product of the two subspace lattices. Furthermore, we give a description of the subspace lattice L ⊗ M and with the help of a result of Harrison in [3] we prove that it is reflexive. We also show that the lattice tensor product formula holds for any Arveson algebra of L and alg M.