Noncommutative Geometry for Symmetric Non-Self-Adjoint Operators

Original · EN

We introduce the notion of a pre-spectral triple, which is a generalisation of a spectral triple (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 Rᵈ, d ≥ 2.