Publication Details

Carey, A. L., Phillips, J. & Rennie, A. C. (2010). Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras On. Journal of K-theory, 6 (2), 339-380.


This paper presents, by example, an index theory appropriate to algebras without trace. Whilst we work exclusively with the Cuntz algebras the exposition is designed to indicate how to develop a general theory. Our main result is an index theorem (formulated in terms of spectral flow) using a twisted cyclic cocycle where the twisting comes from the modular automorphism group for the canonical gauge action on the Cuntz algebra. We introduce a modified K1-group of the Cuntz algebra so as to pair with this twisted cocycle. As a corollary we obtain a noncommutative geometry interpretation for Araki's notion of relative entropy in this example. We also note the connection of this example to the theory of noncommutative manifolds. Contents



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