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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.

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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.

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