Tensor products of σ-weakly closed nest algebra submodules

In this paper we prove that for any unital sigma-weakly closed algebra A which is sigma-weakly generated by finite-rank operators in A, every sigma-weakly closed A-submodule has Property S-sigma. In the case of nest algebras, if L-1,..., L-n are nests, we obtain the following n-fold tensor product formula: mu phi 1 (circle times) over bar...(circle times) over bar mu(phi n) = mu(phi 1 circle times...circle times phi n), where each mu(phi i) is the sigma-weakly closed AlgLi-submodule determined by an order homomorphism phi(i) from L-i into itself.