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In this article, we aim to provide a satisfactory algebraic description ofthe set of affiliated operators for von Neumann algebras. Let $\mathscr{M}$ bea 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 $\ker(B)\subseteq\ker(A)$ ,where $(\cdot)^{\dagger}$ denotes the Kaufman inverse. We show that$\mathscr{M}_{\text{aff}}$ is closed under product, sum, Kaufman-inverse andadjoint; Moreover, the above quotient representation of an operator in$\mathscr{M}_{\text{aff}}$ is essentially unique. Using a novel, simple, andcomplete algebraic characterization of closed operators in terms of boundedoperators, we observe that the Murray-von Neumann affiliated operators for$\mathscr{M}$ are precisely the closed operators in $\mathscr{M}_{\text{aff}}$.Let $\Phi$ be a unital normal homomorphism between represented von Neumannalgebras $(\mathscr{M}; \mathcal{H})$ and $(\mathscr{N}; \mathcal{K})$. Withthe help of 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 respects sum, product, Kaufman-inverse, andadjoint. 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 propertiessuch as being symmetric, or positive, or accretive, or sectorial, orself-adjoint, or normal, and also preserves the Friedrichs and Krein-vonNeumann extensions of densely-defined closed positive operators. As a proof ofconcept, we transfer some well-known results about closed unbounded operatorsto the setting of closed affiliated operators for properly infinite von Neumannalgebras, via `abstract nonsense'.

Algebraic aspects and functoriality of the set of affiliated operators