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

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 Added by Fumio Hiai
 Publication date 2021
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and research's language is English




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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 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 Haagerups $L^p$-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.



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77 - Fumio Hiai 2020
A breakthrough took place in the von Neumann algebra theory when the Tomita-Takesaki theory was established around 1970. Since then, many important issues in the theory were developed through 1970s by Araki, Connes, Haagerup, Takesaki and others, which are already very classics of the von Neumann algebra theory. Nevertheless, it seems still difficult for beginners to access them, though a few big volumes on the theory are available. These lecture notes are delivered as an intensive course in 2019, April at Department of Mathematical Analysis, Budapest University of Technology and Economics. The course was aimed at giving a fast track study of those main classics of the theory, from which people gain an enough background knowledge so that they can consult suitable volumes when more details are needed.
Given a type I von Neumann algebra $M$ with a faithful normal semi-finite trace $tau,$ let $S_0(M, tau)$ be the algebra of all $tau$-compact operators affiliated with $M.$ We give a complete description of all derivations on the algebra $S_0(M, tau).$ In particular, we prove that if $M$ is of type I$_infty$ then every derivation on $S_0(M, tau)$ is spatial.
Given a von Neumann algebra $M$ denote by $S(M)$ and $LS(M)$ respectively the algebras of all measurable and locally measurable operators affiliated with $M.$ For a faithful normal semi-finite trace $tau$ on $M$ let $S(M, tau)$ (resp. $S_0(M, tau)$) be the algebra of all $tau$-measurable (resp. $tau$-compact) operators from $S(M).$ We give a complete description of all derivations on the above algebras of operators in the case of type I von Neumann algebra $M.$ In particular, we prove that if $M$ is of type I$_infty$ then every derivation on $LS(M)$ (resp. $S(M)$ and $S(M,tau)$) is inner, and each derivation on $S_0(M, tau)$ is spatial and implemented by an element from $S(M, tau).$
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279 - Lu Cui , Linzhe Huang , Wenming Wu 2021
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