Connes proved in his beautiful paper "Compact metric spaces, Fredholm modules, and hyperfiniteness" published in 1989 that if $(A,H,D)$ is a finitely summable spectral triple with a unital $C^*$-algebra $A$, then $A$ must have a tracial state. Here is the question: Is there an analogue of this result for non-unital spectral triples (for which $A$ is a non-unital $C^*$-algebra and $D$ has locally compact resolvents)? What are the main obstructions for existence of finitely summable spectral triples?


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