Noncommutative Geometry for Symmetric Non-Self-Adjoint Operators

TL;DR

This paper extends noncommutative geometry by introducing pre-spectral triples that relax self-adjointness, enabling analysis of boundary conditions and deriving a Hochschild character theorem in this broader context.

Contribution

It introduces pre-spectral triples, generalizing spectral triples to include non-self-adjoint operators, and applies this to boundary value problems in noncommutative geometry.

Findings

Established the Hochschild character theorem for pre-spectral triples.

Analyzed Dirac operators with Dirichlet boundary conditions.

Extended noncommutative geometry to noncompact spaces with boundary.

Abstract

We introduce the notion of a pre-spectral triple, which is a generalisation of a spectral triple $(\mathcal{A}, H, D)$ where $D$ is no longer required to be self-adjoint, but closed and symmetric. Despite having weaker assumptions, pre-spectral triples allow us to introduce noncompact noncommutative geometry with boundary. In particular, we derive the Hochschild character theorem in this setting. We give a detailed study of Dirac operators with Dirichlet boundary conditions on open subsets of $\mathbb{R}^d$, $d \geq 2$.