Boundaries, spectral triples and $K$-homology

Abstract

This paper extends the notion of a spectral triple to a relative spectral triple, an unbounded analogue of a relative Fredholm module for an ideal $J◃A$. Examples include manifolds with boundary, manifolds with conical singularities, dimension drop algebras, $\theta$-deformations and Cuntz–Pimsner algebras of vector bundles.

The bounded transform of a relative spectral triple is a relative Fredholm module, making the image of a relative spectral triple under the boundary mapping in $K$-homology easy to compute. We introduce an additional operator called a Clifford normal with which a relative spectral triple can be doubled into a spectral triple.

The Clifford normal also provides a boundary Hilbert space, a representation of the quotient algebra, a boundary Dirac operator and an analogue of the Calderon projection. In the examples this data does assemble to give a boundary spectral triple, though we can not prove this in general.

When we do obtain a boundary spectral triple, we provide sufficient conditions for the boundary triple to represent the $K$-homological boundary. Thus we abstract the proof of Baum–Douglas–Taylor's "boundary of Dirac is Dirac on the boundary" theorem into the realm of non-commutative geometry.