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
HTML articles powered by AMS MathViewer

by De Guang Han PDF

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

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 $\mathcal {M}$.

References

Erik Christensen, Derivations of nest algebras, Math. Ann. 229 (1977), no. 2, 155–161. MR 448110, DOI 10.1007/BF01351601
J. A. Erdos and S. C. Power, Weakly closed ideals of nest algebras, J. Operator Theory 7 (1982), no. 2, 219–235. MR 658610
Jon Kraus and David R. Larson, Reflexivity and distance formulae, Proc. London Math. Soc. (3) 53 (1986), no. 2, 340–356. MR 850224, DOI 10.1112/plms/s3-53.2.340
Cecelia Laurie and W. E. Longstaff, A note on rank-one operators in reflexive algebras, Proc. Amer. Math. Soc. 89 (1983), no. 2, 293–297. MR 712641, DOI 10.1090/S0002-9939-1983-0712641-2