A bounded transform approach to self-adjoint operators: Functional calculus and affiliated von Neumann algebras

TL;DR

This paper introduces a new approach to spectral theory and functional calculus for unbounded self-adjoint operators using bounded transforms, connecting them with von Neumann algebras via multiplier algebras.

Contribution

It develops a bounded transform framework based on Woronowicz's ideas, simplifying the treatment of unbounded operators and their affiliation with von Neumann algebras.

Findings

Establishes a bijective correspondence between closed operators and pure contractions.

Provides a simplified account of the affiliation relation between von Neumann algebras and self-adjoint operators.

Reformulates spectral theory using multiplier algebras and bounded transforms.

Abstract

Spectral theory and functional calculus for unbounded self-adjoint operators on a Hilbert space are usually treated through von Neumann's Cayley transform. Based on ideas of Woronowicz, we redevelop this theory from the point of view of multiplier algebras and the so-called bounded transform (which establishes a bijective correspondence between closed operators and pure contractions). This also leads to a simple account of the affiliation relation between von Neumann algebras and self-adjoint operators.