In recent works [G. A. Bagheri-Bardi, A. Elyaspour and G. H. Esslamzadeh, Wold-type decompositions in Baer ∗\ast-rings, Linear Algebra Appl. 539 2018, 117–133] and [G. A. Bagheri-Bardi, A. Elyaspour and G. H. Esslamzadeh, The role of algebraic structure in the invariant subspace theory, Linear Algebra Appl. 583 2019, 102–118], the algebraic analogues of the three major decomposition theorems of Wold, Nagy–Foiaş–Langer and Halmos–Wallen were established in the larger category of Baer *{*}-rings. The results have their versions for commuting pairs in von Neumann algebras. In the corresponding proofs, both norm and weak operator topologies are heavily involved. In this work, ignoring topological structures, we give an algebraic approach to obtain them in Baer *{*}-rings.

中文翻译：

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代数结构在 Wold 型分解中的作用

在最近的作品中 [GA Bagheri-Bardi、A. Elyaspour 和 GH Esslamzadeh，Baer ∗\ast-rings 中的 Wold 型分解，线性代数应用。539 2018, 117–133] 和 [GA Bagheri-Bardi, A. Elyaspour 和 GH Esslamzadeh，代数结构在不变子空间理论中的作用，线性代数应用。583 2019, 102-118]，Wold、Nagy-Foiaş-Langer 和 Halmos-Wallen 的三个主要分解定理的代数类似物是在更大的 Baer *{*}-环类中建立的。结果对于冯诺依曼代数中的交换对有其版本。在相应的证明中，范数和弱算子拓扑都被大量涉及。在这项工作中，忽略拓扑结构，我们给出了在 Baer *{*} 环中获得它们的代数方法。