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

Abstract

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 [Formula: see text]-measurable operators (affiliated with a semifinite von Neumann algebra equipped with a trace [Formula: see text]), (ii) positive elements in Haagerup’s [Formula: see text]-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) [Formula: see text]-spaces.