Orthogonal Decompositions in Hilbert A-Modules

Pre-Hilbert A-Modules are a natural generalization of inner product spaces in which the scalars are allowed to be from an arbitrary algebra. In this perspective, submodules are the generalization of vector subspaces. The notion of orthogonality generalizes in an obvious way too. In this paper, we provide necessary and sufficient topological conditions for a submodule to be orthogonally complemented. We present three applications of our results. In the most important one, we obtain the Kunita-Watanabe decomposition for conditionally square-integrable martingales as an orthogonal decomposition result carried out in an opportune pre-Hilbert A-module. Second, we show that a version of Stricker's Lemma can be also derived as a corollary of our results. Finally, we provide a version of the Koopman-von Neumann decomposition theorem for a specific pre-Hilbert module which is useful in Ergodic Theory.