Constructing KMS states from infinite-dimensional spectral triples

Abstract

We construct KMS-states from Li 1 -summable semifinite spectral triples and show that in several important examples the construction coincides with well-known direct constructions of KMS-states for naturally defined flows. Under further summability assumptions the constructed KMS-state can be computed in terms of Dixmier traces. For closed manifolds, we recover the ordinary Lebesgue integral. For Cuntz-Pimsner algebras with their gauge flow, the construction produces KMS-states from traces on the coefficient algebra and recovers the Laca-Neshveyev correspondence. For a discrete group acting on its Stone-Cech boundary, we recover the Patterson-Sullivan measures on the Stone-Čech boundary for a flow defined from the Radon-Nikodym cocycle. *

Untwisting twisted spectral triples

“…Our two main results indicate that the analytic aspect of index theory for finitely summable twisted spectral triples is quite involved. The proofs of these results indicate that the appropriate index theory is in fact closely related to the index theory for Li 1 -summable spectral triples (recently studied in [30]).…”

“…The proposition follows from the definitions, see [30], because Corollary 1.20. Let (A, X B , D) be a triple containing the following information:…”

“…Now a Dom(|D|) ⊂ Dom(|D|) and Lemma 1.15 applied to ∆ = |D| gives that [log |D|, a] is bounded, so we are done. The proposition follows from the definitions, see [30], because…”

“…Our first general result states that functional calculus using the C 1 -function sgnlog : R → R, x → sgn(x) log(1 + |x|), can be used to turn both twisted and higher order spectral triples into ordinary spectral triples without changing the K-homology class (for details on the twisted case, see Theorem 1 below). This logarithmic dampening has been used before in specific examples [7,24,26,30,44,46], and in Section 1 of the present paper we formalise the procedure.…”

“…Our first general result states that functional calculus using the C 1 -function sgnlog : R → R, x → sgn(x) log(1 + |x|), can be used to turn both twisted and higher order spectral triples into ordinary spectral triples without changing the K-homology class (for details on the twisted case, see Theorem 1 below). This logarithmic dampening has been used before in specific examples [7,24,26,30,44,46], and in Section 1 of the present paper we formalise the procedure. Thus, if a twisted or higher order spectral triple encodes non-trivial index theoretic data, then the same information can be recovered from an ordinary spectral triple, constructed through a well-defined procedure, albeit possibly less geometric than the original twisted spectral triple.…”