Noncommutative Atiyah-Patodi-Singer boundary conditions and index pairings in KK-theory

A. L. Carey; J. Phillips; A. Rennie

doi:10.1515/CRELLE.2010.045

Noncommutative Atiyah-Patodi-Singer boundary conditions and index pairings in KK-theory

A. L. Carey, J. Phillips, A. Rennie

Research output: Contribution to journal › Article › peer-review

8 Citations (Scopus)

Abstract

We investigate an extension of ideas of Atiyah-Patodi-Singer (APS) to a noncommutative geometry setting framed in terms of Kasparov modules. We use a mapping cone construction to relate odd index pairings to even index pairings with APS boundary conditions in the setting of KK-theory, generalising the commutative theory. We find that Cuntz-Krieger systems provide a natural class of examples for our construction and the index pairings coming from APS boundary conditions yield complete K-theoretic information about certain graph C□-algebras.

Original language	English
Pages (from-to)	59-109
Number of pages	51
Journal	Journal fur die Reine und Angewandte Mathematik
Issue number	643
DOIs	https://doi.org/10.1515/CRELLE.2010.045
Publication status	Published - Jun 2010

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abstract = "We investigate an extension of ideas of Atiyah-Patodi-Singer (APS) to a noncommutative geometry setting framed in terms of Kasparov modules. We use a mapping cone construction to relate odd index pairings to even index pairings with APS boundary conditions in the setting of KK-theory, generalising the commutative theory. We find that Cuntz-Krieger systems provide a natural class of examples for our construction and the index pairings coming from APS boundary conditions yield complete K-theoretic information about certain graph C□-algebras.",

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AB - We investigate an extension of ideas of Atiyah-Patodi-Singer (APS) to a noncommutative geometry setting framed in terms of Kasparov modules. We use a mapping cone construction to relate odd index pairings to even index pairings with APS boundary conditions in the setting of KK-theory, generalising the commutative theory. We find that Cuntz-Krieger systems provide a natural class of examples for our construction and the index pairings coming from APS boundary conditions yield complete K-theoretic information about certain graph C□-algebras.

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Noncommutative Atiyah-Patodi-Singer boundary conditions and index pairings in KK-theory

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