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 $\mathcal{M}$ be a von Neumann algebra acting on a Hilbert space $\mathcal{H}$, and let ${\mathcal{M}}_{\text{aff}}$ denote the set of unbounded operators of the form $T=A{B}^{†}$ for $A,B\in \mathcal{M}$ with $\ker (B)\subseteq \ker (A)$ , where $(\cdot {)}^{†}$ denotes the Kaufman inverse. We show that ${\mathcal{M}}_{\text{aff}}$ is closed under product, sum, Kaufman-inverse and adjoint, and has the structure of a right near-semiring; Moreover, the above quotient representation of an operator in ${\mathcal{M}}_{\text{aff}}$ is essentially unique. The Murray-von Neumann affiliated operators for $\mathcal{M}$ turn out to be precisely the closed operators in ${\mathcal{M}}_{\text{aff}}$. Let $\Phi$ be a unital normal homomorphism between represented von Neumann algebras $(\mathcal{M};\mathcal{H})$ and $(\mathcal{N};\mathcal{K})$. With the help of the quotient representation, we obtain a canonical extension of $\Phi$ to a mapping ${\Phi }_{\text{aff}}:{\mathcal{M}}_{\text{aff}}\to {\mathcal{N}}_{\text{aff}}$ which respects sum, product, Kaufman-inverse, and adjoint. Thus ${\mathcal{M}}_{\text{aff}}$ is intrinsically associated with $\mathcal{M}$ and transforms functorially as we change representations of $\mathcal{M}$. 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'.

39 pages. Accepted for publication in IMRN