On Algebraic Aspects and Functoriality of the Set of Unbounded Operators Affiliated with a von Neumann Algebra

Abstract

Abstract. In quantum theory, physically meaningful observables such as position and momentum are modeled by unbounded self-adjoint operators on Hilbert spaces. To study such operators in a rigorous mathematical framework, Murray and von Neumann introduced the concept of affiliation with a von Neumann algebra, giving rise to what are now known as Murray–von Neumann affiliated operators. For a single (unbounded) self-adjoint operator, its spectral projections span an abelian von Neumann algebra to which the operator is affiliated. If we want a common framework to study families of (unbounded) self-adjoint operators which do not necessarily commute with each other, it is instructive to study the notion of affiliation for general von Neumann algebras. Unfortunately, the set of affiliated operators has so far not been realized as a familiar algebraic structure that would allow for a systematic study. For example, they are not closed under natural algebraic operations such as addition and multiplication. Moreover, their behavior under morphisms of von Neumann algebras is not immediately evident from the classical definition. This thesis addresses these limitations by introducing a new, intrinsically defined class of unbounded operators affiliated to a von Neumann algebra, which generalizes the classical notion while retaining its essential features. The proposed class forms a near-semiring, is functorial with respect to unital normal ∗-homomorphisms of von Neumann algebras, and contains all MvN-affiliated operators. As a key application, we show that the classical construction of MvN-affiliated operators itself becomes functorial within this broader framework. We also explore connections with measurable and locally measurable operator algebras, and clarify the subtleties in their functorial behavior. In the later chapters, we revisit the Krein–von Neumann extension theory and establish the functoriality of both the Krein and Friedrich extensions for positive symmetric operators affiliated with a given von Neumann algebra. Additionally, we extend our affiliation framework beyond von Neumann algebras to monotone σ-complete C∗-algebras, enabling a more general theory with potential applications to noncommutative geometry and mathematical physics. The results developed in this thesis provide a robust algebraic and categorical foundation for studying unbounded affiliated operators in operator algebras, and open the door to new directions in the analysis of infinite quantum systems and beyond.