The Kubo–Ando theory deals with connections for positive bounded operators. On the other hand, in various analysis related to von Neumann algebras, it is impossible to avoid unbounded operators. In this paper, we try to extend a notion of connections to cover various classes of positive unbounded operators (or unbounded objects such as positive forms and weights) appearing naturally in the setting of von Neumann algebras, and we must keep all the expected properties maintained. This generalization is carried out for the following classes: (i) positive τ-measurable operators (affiliated with a semifinite von Neumann algebra equipped with a trace τ), (ii) positive elements in Haagerup’s Lp-spaces and (iii) semifinite normal weights on a von Neumann algebra. Investigation on these generalizations requires some analysis (such as certain upper semi-continuity) on decreasing sequences in various classes. Several results in this direction are proved, which may be of independent interest. Ando studied Lebesgue decomposition for positive bounded operators by making use of parallel sums. Here, such decomposition is obtained in the setting of noncommutative (Hilsum) Lp-spaces.

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无界算子的连接和一些相关主题：冯诺依曼代数案例

Kubo-Ando 理论处理正有界算子的连接。另一方面，在与冯诺依曼代数相关的各种分析中，不可能避免无界算子。在本文中，我们尝试扩展连接的概念以涵盖在冯诺依曼代数设置中自然出现的各种类型的正无界算子（或正数形式和权重等无界对象），并且我们必须保持所有预期的性质. 这种概括是针对以下类别进行的：（i）积极的τ- 可测量算子（附属于配备迹线的半有限冯诺依曼代数τ), (ii) Haagerup 的积极因素大号p-空间和 (iii) 冯诺依曼代数上的半有限正态权重。对这些概括的研究需要对各种类别的递减序列进行一些分析（例如某些上半连续性）。证明了这个方向的几个结果，这些结果可能具有独立的意义。Ando 通过使用并行求和研究了正有界算子的 Lebesgue 分解。在这里，这样的分解是在非交换（Hilsum）的设置中获得的大号p-空格。