A remark on trace properties of K-cycles

Abstract

AREMARKONTRACEPROPERTIESOFK-CYCLES.Fabio Cipriani1DanieleGuido2and Sergio Scarlatti(1)Mathematical Department, University ofNottingham,University Park NG7-2RD, Nottingham UK(2)Dipartimento di Matematica, Universita di Roma \Tor VergataVia della Ricerca Scienti ca, I{00133 Roma, Italia.Abstract.Inthispap erwediscusstraceprop ertiesofd+-summableK-cyclescon-sidered byA.Connesin[5].Morepreciselywegivapro ofoftracetheoremonthealgebraAofaK{cyclestatedin[5],namelyweshothatnaturalfunctionalonAisatracefunctional.Thenwediscuss whetherthisfunctionalgivesonthewholeuniversal graded di erential algebra (A).Ontheonehand weprove thatregularity conditions onK-cycles considered in[5] imply thetraceprop erty on(A).Ontheotherhand,byconstructinganexplicitcounterexample,weremarkthatsoleK-cycleassumption is notsucient for such a prop erty tohold.Supp orted by H.C.M. EC-grantSupp orted in part by MURST and CNR-GNAFA.E-mail: [e-mail], [e-mail], [e-mail]1