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

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Journal fur die Reine und Angewandte Mathematik, ISSN: 0075-4102, Vol: 2010, Issue: 643, Page: 59-109

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Carey, Alan L; Phillips, John; Rennie, Adam C
Walter de Gruyter GmbH
Mathematics; Engineering; Science and Technology Studies
article description
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. © Walter de Gruyter Berlin . New York 2010.