  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 $\ker(B)\subseteq\ker(A)$ ,
where $(\cdot)^{\dagger}$ denotes the Kaufman inverse. We show that
$\mathscr{M}_{\text{aff}}$ is closed under product, sum, Kaufman-inverse and
adjoint; Moreover, the above quotient representation of an operator in
$\mathscr{M}_{\text{aff}}$ is essentially unique. Using a novel, simple, and
complete algebraic characterization of closed operators in terms of bounded
operators, 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 Neumann
algebras $(\mathscr{M}; \mathcal{H})$ and $(\mathscr{N}; \mathcal{K})$. With
the 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, and
adjoint. 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'.
