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. We provide necessary and sufficient topological conditions for a submodule to be orthogonally complemented. Then, we present four applications of our results. The most important ones are Doob's and Kunita–Watanabe's decompositions for conditionally square-integrable processes. They are obtained as orthogonal decomposition results 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.