Abstract. Let ${\cal A}$ be a reflexive operator algebra acting on a Hilbert space ${\cal H}$, ${\cal U}$ be a reflexive ${\cal A}$-module, and ${\cal U}_\perp $ be the preannihilator of ${\cal U}$. Suppose $\phi $ is an order homomorphism of Lat${\cal A}$ determining ${\cal U}$, that is ${\cal U}={\cal U}_\phi := \{T\in{\cal B} ({\cal H}): \phi(E)^\perp TE = 0, \forall E\in\mathop{\rm Lat}{\cal A}\} $. In this paper, it is proved that $\phi_\sim(E) = \left[{\cal U}_\perp E\right ] =\left[{\cal U}_{\phi_\sim } E\right ]$ for each $E\in\mathop{\rm Lat}{\cal A}$, where $\phi_\sim(E)$ is defined as $\vee\{F\in\mathop{\rm Lat}{\cal A}: \phi(F)\not\supseteq E \} $. We also characterize the invariant subspaces of ${\cal U}_\perp $ in terms of order homomorphisms of Lat${\cal A}$, and show that if Lat${\cal A}$ is a nest then Lat${\cal A}_\perp =\mathop{\rm Lat}{\cal A}$ if and only if Lat${\cal A}$ is maximal. Moreover, we investigate the relationships between reflexive modules and their preannihilators. If ${\cal A}$ is additionally $\sigma $--weakly generated by rank one operators, a necessary and sufficient condition for an order homomorphism to be the least one determining ${\cal U}$ is given.

AMS Subject Classification (1991): 47L05, 47L35, 47L75

Keyword(s): Reflexivity, module, preannihilator, invariant subspace, order homomorphism

Received September 26, 2000, and in final form July 13, 2001. (Registered under 2850/2009.)