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

2021/01/04 by Fumio Hiai, Hiai, Fumio, Hideki Kosaki +1
Mathematics · #Advanced Operator Algebra Research #Advanced Banach Space Theory #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2101.01176

Abstract

The Kubo-Ando theory deals with connections for positive bounded operators.\nOn the other hand, in various analysis related to von Neumann algebras it is\nimpossible to avoid unbounded operators. In this article we try to extend a\nnotion of connections to cover various classes of positive unbounded operators\n(or unbounded objects such as positive forms and weights) appearing naturally\nin the setting of von Neumann algebras, and we must keep all the expected\nproperties maintained. This generalization is carried out for the following\nclasses: (i) positive \τ-measurable operators (affiliated with a\nsemi-finite von Neumann algebra equipped with a trace \τ), (ii) positive\nelements in Haagerup's L^p-spaces, (iii) semi-finite normal weights on a von\nNeumann algebra. Investigation on these generalizations requires some analysis\n(such as certain upper semi-continuity) on decreasing sequences in various\nclasses. Several results in this direction are proved, which may be of\nindependent interest. Ando studied Lebesgue decomposition for positive bounded\noperators by making use of parallel sums. Here, such decomposition is obtained\nin the setting of non-commutative (Hilsum) L^p-spaces.\n

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