On 𝒜-submodules for reflexive operator algebras

Han, De Guang

doi:10.1090/S0002-9939-1988-0969048-7

On $\mathcal {A}$-submodules for reflexive operator algebras
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by De Guang Han

Proc. Amer. Math. Soc. 104 (1988), 1067-1070

DOI: https://doi.org/10.1090/S0002-9939-1988-0969048-7

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Abstract:

In [2] the authors described all weakly closed $\mathcal {A}$-submodules of $L\left ( H \right )$ for a nest algebra $\mathcal {A}$ in terms of order homomorphisms of Lat $\mathcal {A}$. In this paper we prove that for any reflexive algebra $\mathcal {A}$ which is $\sigma$-weakly generated by rank-one operators in $\mathcal {A}$, every $\sigma$-weakly closed $\mathcal {A}$-submodule can be characterized by an order homomorphism of Lat $\mathcal {A}$. In the case when $\mathcal {A}$ is a reflexive algebra with a completely distributive subspace lattice and $\mathcal {M}$ is a $\sigma$-weakly closed ideal of $\mathcal {A}$, we obtain necessary and sufficient conditions for the commutant of $\mathcal {A}$ modulo $\mathcal {M}$ to be equal to AlgLat $\mathcal {M}$.

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