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

TL;DR

This paper extends the concept of operator connections from bounded to unbounded operators within von Neumann algebras, developing new theoretical tools for analyzing positive unbounded operators and their decompositions.

Contribution

It introduces a generalized notion of connections for various classes of positive unbounded operators in von Neumann algebras, preserving key properties and establishing new decomposition results.

Findings

Extended connections to positive unbounded operators in von Neumann algebras.

Proved decomposition results in non-commutative L^p-spaces.

Analyzed properties like upper semi-continuity for these classes.

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 article 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 $\tau$-measurable operators (affiliated with a semi-finite von Neumann algebra equipped with a trace $\tau$), (ii) positive elements in Haagerup's $L^p$-spaces, (iii) semi-finite 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 non-commutative (Hilsum) $L^p$-spaces.