Algebraic Aspects and Functoriality of the Set of Affiliated Operators

Article Type

Research Article

Publication Title

International Mathematics Research Notices

Abstract

In this article, we aim to provide a satisfactory algebraic description of the set of affiliated operators for von Neumann algebras. Let be a von Neumann algebra acting on a Hilbert space, and let denote the set of unbounded operators of the form for with, where denotes the Kaufman inverse. We show that is closed under sum, product, Kaufman-inverse, and adjoint, and has the structure of a (right) near-semiring. Moreover, the above quotient representation of an operator in is essentially unique. Thus, we may view as the multiplicative monoid of unbounded operators on generated by and. We further show that our definition of affiliation, as reflected in, subsumes the traditional one. Let be a unital normal ∗-homomorphism between represented von Neumann algebras and. Using the quotient representation, we obtain a canonical extension of to a mapping which is a near-semiring homomorphism that respects Kaufman-inverse and adjoint; in addition, respects Murray-von Neumann affiliation of operators and also respects strong sum and strong product. Thus, is intrinsically associated with and transforms functorially as we change representations of. Furthermore, preserves operator properties such as being symmetric, or positive, or accretive, or sectorial, or self-adjoint, or normal, and also preserves the Friedrichs and Krein-von Neumann extensions of densely defined closed positive operators. As a proof of concept, we transfer some well-known results about closed unbounded operators to the setting of closed affiliated operators for properly infinite von Neumann algebras, via "abstract nonsense".

First Page

13525

Last Page

13562

DOI

10.1093/imrn/rnae203

Publication Date

11-1-2024

Comments

Open Access; Green Open Access

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