Tensor products of $\sigma $-weakly closed nest algebra submodules

Abstract

In this paper we prove that for any unital cr-weakly closed algebra A which is σ-weakly generated by finite-rank operators in A, every σ-weakly closed A-submodule has Property S σ . In the case of nest algebras, if L 1 ,..., L n are nests, we obtain the following n-fold tensor product formula: u Φ1 ⊗...⊗u Φn = u Φ1⊗...⊗Φn , where each u Φi is the σ-weakly closed AlgL i -submodule determined by an order homomorphism Φ i from L i into itself.