Skip to main content
placeholder image

Boundaries, spectral triples and K-homology

Journal Article


Download full-text (Open Access)

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 G A. Examples include manifolds with boundary, manifolds with conical singularities, dimension drop algebras, -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.

Authors


  •   Forsyth, Iain G. (external author)
  •   GOFFENG, MAGNUS (external author)
  •   Mesland, Bram (external author)
  •   Rennie, Adam C.

Publication Date


  • 2019

Citation


  • Forsyth, I., Goffeng, M., Mesland, B. & Rennie, A. (2019). Boundaries, spectral triples and K-homology. Journal of Noncommutative Geometry, 13 (2), 407-472.

Scopus Eid


  • 2-s2.0-85070505143

Ro Full-text Url


  • https://ro.uow.edu.au/cgi/viewcontent.cgi?article=4127&context=eispapers1

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers1/3108

Number Of Pages


  • 65

Start Page


  • 407

End Page


  • 472

Volume


  • 13

Issue


  • 2

Place Of Publication


  • Switzerland

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 G A. Examples include manifolds with boundary, manifolds with conical singularities, dimension drop algebras, -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.

Authors


  •   Forsyth, Iain G. (external author)
  •   GOFFENG, MAGNUS (external author)
  •   Mesland, Bram (external author)
  •   Rennie, Adam C.

Publication Date


  • 2019

Citation


  • Forsyth, I., Goffeng, M., Mesland, B. & Rennie, A. (2019). Boundaries, spectral triples and K-homology. Journal of Noncommutative Geometry, 13 (2), 407-472.

Scopus Eid


  • 2-s2.0-85070505143

Ro Full-text Url


  • https://ro.uow.edu.au/cgi/viewcontent.cgi?article=4127&context=eispapers1

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers1/3108

Number Of Pages


  • 65

Start Page


  • 407

End Page


  • 472

Volume


  • 13

Issue


  • 2

Place Of Publication


  • Switzerland