Algebraic Aspects and Functoriality of the Set of Affiliated Operators

In this article, we aim to provide a satisfactory algebraic description of the set of affiliated operators for von Neumann algebras. Let M be a von Neumann algebra acting on a Hilbert space H, and let Maff denote the set of unbounded operators of the form T = AB for A, B ∈ M with null (B) ⊆ null (A), where (· ) denotes the Kaufman inverse. We show that Maff 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 Maff is essentially unique. Thus, we may view Maff as the multiplicative monoid of unbounded operators on H generated by M and M. We further show that our definition of affiliation, as reflected in Maff, subsumes the traditional one. Let Φ be a unital normal *-homomorphism between represented von Neumann algebras (M; H) and (N; K). Using the quotient representation, we obtain a canonical extension of Φ to a mapping Φ aff: Maff → Naff which is a near-semiring homomorphism that respects Kaufman-inverse and adjoint; in addition, Φ aff respects Murray–von Neumann affiliation of operators and also respects strong sum and strong product. Thus, Maff is intrinsically associated with M and transforms functorially as we change representations of M. Furthermore, Φ aff 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”.