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

Abstract:

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: \[ \mathcal U_{\phi _{1}}{\overline {\otimes }}\cdots {\overline {\otimes }} \mathcal U_{\phi _{n}}= \mathcal U_{\phi _{1}\otimes \cdots \otimes \phi _{n}},\] 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.