Algebraic Aspects and Functoriality of the Set of Affiliated Operators

Abstract

Closed unbounded operators on Hilbert spaces (such as differential operators) play an important role in applications of operator theory to the theory of differential equations and dynamical systems. The set of affiliated operators, $\mathscr{M}_{\text{aff}}$, for a von Neumann algebra, $\mathscr{M}$, acting on a Hilbert space $\mathcal{H}$ consists of a certain class of closed operators on $\mathcal{H}$ associated with $\mathscr{M}$. When $\mathscr{M}$ is a finite von Neumann algebra, it is well-known that $\mathscr{M}_{\text{aff}}$ may be viewed as the Ore localization of $\mathscr{M}$ relative to the multiplicative subset of non-zero-divisors. So far, the algebraic structure of $\mathscr{M}_{\text{aff}}$ for an arbitrary von Neumann algebra $\mathscr{M}$ has remained mysterious. In this talk, we will unravel some of this mystery by viewing $\mathscr{M}_{\text{aff}}$ as the multiplicative monoid of unbounded operators on $\mathcal{H}$ generated by $\mathscr{M}$ and $\mathscr{M} ^{\dagger}$ where $(\cdot)^{\dagger}$ denotes the Kaufman-inverse. We will see that our definition of affiliation subsumes the traditional one due to Murray and von Neumann. Our main goal in this talk is to highlight the conceptual and technical simplifications that the new definition provides. For instance, the intrinsic nature of $\mathscr{M}_{\text{aff}}$ (independent of the representation of $\mathscr{M}$ on Hilbert space) will become apparent after showing the functorial nature of the construction $\mathscr{M} \mapsto \mathscr{M}_{\text{aff}}$. This talk is based on joint work with my PhD student, Indrajit Ghosh.