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

In this paper we prove that for any unital $\sigma$-weakly closed algebra $\mathcal A$ which is $\sigma$-weakly generated by finite-rank operators in $\mathcal A$, every $\sigma$-weakly closed $\mathcal A$-submodule has $Property\; S_{\sigma}$. In the case of nest algebras, if $\mathcal L_{1},\cdots,\mathcal L_{n}$ are nests, we obtain the following $n$-fold tensor product formula:

where each $\mathcal U_{\phi_{i}}$ is the $\sigma$-weakly closed Alg $\mathcal L_{i}$-submodule determined by an order homomorphism $\phi_{i}$ from $\mathcal L_{i}$ into itself.