On A-Submodules for Reflexive Operator Algebras

Abstract

In [2] the authors described all weakly closed _W-submodules of L(H) for a nest algebra v in terms of order homomorphisms of Lat-'. In this paper we prove that for any reflexive algebra v which is a-weakly generated by rank-one operators in X, every a-weakly closed s/-submodule can be characterized by an order homomorphism of Lat-W. In the case when v is a reflexive algebra with a completely distributive subspace lattice and / is a a-weakly closed ideal of X, we obtain necessary and sufficient conditions for the commutant of v modulo X to be equal to AlgLatf. Let H be a complex Hilbert space, and let L(H) be the set of all bounded linear operators on H. The terminology and notation of this paper concerning nest algebras and reflexive subspaces of L(H) may be found in [3]. Let v be a reflexive subalgebra of L(H). Suppose that E is an order homomorphism of Lat _W into itself (i.e. E < F implies E < F), where Lat &' is the set of all invariant projections for -W. Then the set //'X = {T c L(H): (I-E)TE = 0 for all E E Lat -W} is clearly a weakly closed -W -subrnodule of L(H). J. A. Erdos and S. C. Power in [2] proved that any weakly closed -W-submodule of L(H) for a nest algebra v is of the above form. Here we prove that this is also true for any reflexive algebra v which is a-weakly generated by rank-one operators in X. The following result is due to J. Kraus and D. R. Larson [3]. THEOREM 1. Let v be a unital a-weakly closed algebra which is a-weakly generated by rank-one operators in -W. Then every a-weakly closed left or right module of is reflexive. THEOREM 2. Let v be as in the above theorem, and let X be a a-weakly closed v -submodule of L (H). Then X' has the form X = {T c L(H): (I E)TE = 0 for all E c Lat-'}, where E f-* E is some order homomorphism of Lat-W into itself. PROOF. For any E c Lat-W, let E be the orthogonal projection onto [/'EH] = V{ran(XE): X c X}. Since I is an -'-submodule, E is invariant under -v and clearly E ~-+ E is an order homomorphism. Let -I = {T c L(H): (I E)TE = 0 for all E c Lat-&}. It is obvious that X D X'. Conversely, if T c IV, then (I E)TE = 0, so [TEH] C [EH] = [vfEH] for any E E Lat-'. Now for any Received by the editors March 20, 1987 and, in revised form, December 7, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 47D25, 47D15; Secondary 47B47.