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Abstract
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Following a recent work of authors \cite{BEE}, we establish the algebraic analogs of three major decomposition theorems of Nagy-Foias-Langer, Halmos-Wallen and Fishel in Baer $*$-rings. This approach not only provides purely algebraic and shorter proofs of these decomposition theorems, but also suggests the possibility of eliminating the role of analytic structure in results related to the invariant subspace theory.
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