Algebraic Aspects and Functoriality of the Set of Affiliated Operators

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