  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.
