Journal of Operator Theory

Volume 35, Issue 1, Winter 1996 pp. 179-189.

A remark on trace properties of K-cycles

Authors: Fabio Cipriani (1), Daniele Guido (2) and Sergio Scarlatti (3)
Author institution:(1) Mathematical Department, University of Nottingham, University Park NG7-2RD, Nottingham, UNITED KINGDOM
(2) Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, I-00133 Roma, ITALIA
(3) Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, I-00133 Roma, ITALIA

Summary: In this paper we discuss trace properties of d^+-summable K-cycles considered by A. Connes in [6]. More precisely we give a proof of a trace theorem on the algebra $\mathcal A$ of a K-cycle stated in [6], namely we show that a natural functional on $\mathcal A$ is a trace functional. Then we discuss whether this functional gives a trace on the whole universal graded differential algebra $\Omega(\mathcal A)$. On the one hand we prove that the regularity conditions on K-cycles considered in [6] imply the trace property on $\Omega(\mathcal A)$. 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.

Keywords: Noncommutative geometry, Dirac operator, traces on K-cycles, singular traces.

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