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A remark on trace properties of K-cycles

Fabio Cipriani·Daniele Guido·Sergio Scarlatti
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Abstract

In this paper we discuss trace properties of $d^+$-summable $K$-cycles considered by A.Connes in [\rfr(Conn4)]. More precisely we give a proof of a trace theorem on the algebra $\A$ of a $K$--cycle stated in [\rfr(Conn4)], namely we show that a natural functional on $\A$ is a trace functional. Then we discuss whether this functional gives a trace on the whole universal graded differential algebra $\Q(\A)$. On the one hand we prove that the regularity conditions on $K$-cycles considered in [\rfr(Conn4)] imply the trace property on $\Q(\A)$. On the other hand, by constructing an explicit counterexample, we remark that the sole $K$-cycle assumption is not sufficient for such a property to hold.

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Fabio Cipriani, Daniele Guido, Sergio Scarlatti. 1995-06-14. A remark on trace properties of K-cycles. https://arxiv.org/abs/funct-an/9506003

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