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

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 avoid unbounded In this paper, we try extend a notion of cover classes operators (or objects such as forms and weights) appearing naturally setting must keep all expected properties maintained. This generalization carried out following classes: (i) [Formula: see text]-measurable (affiliated semifinite algebra equipped trace text]), (ii) elements Haagerup’s text]-spaces (iii) normal weights on algebra. Investigation these generalizations requires some (such certain upper semi-continuity) decreasing sequences classes. Several results direction are proved, which may be independent interest. Ando studied Lebesgue decomposition by making use parallel sums. Here, obtained noncommutative (Hilsum) text]-spaces.