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, UniversitÃ di Roma â€œTor Vergataâ€, Via della Ricerca Scientifica, I-00133 Roma, ITALIA

$3$ Dipartimento di Matematica, UniversitÃ di Roma â€œTor Vergataâ€, 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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