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The Kubo-Ando theory deals with connections for positive bounded operators.On the other hand, in various analysis related to von Neumann algebras it isimpossible to avoid unbounded operators. In this article we try to extend anotion of connections to cover various classes of positive unbounded operators(or unbounded objects such as positive forms and weights) appearing naturallyin the setting of von Neumann algebras, and we must keep all the expectedproperties maintained. This generalization is carried out for the followingclasses: (i) positive $\tau$-measurable operators (affiliated with asemi-finite von Neumann algebra equipped with a trace $\tau$), (ii) positiveelements in Haagerup's $L^p$-spaces, (iii) semi-finite normal weights on a vonNeumann algebra. Investigation on these generalizations requires some analysis(such as certain upper semi-continuity) on decreasing sequences in variousclasses. Several results in this direction are proved, which may be ofindependent interest. Ando studied Lebesgue decomposition for positive boundedoperators by making use of parallel sums. Here, such decomposition is obtainedin the setting of non-commutative (Hilsum) $L^p$-spaces.

Connections of unbounded operators and some related topics: von Neumann algebra case