A remark on trace properties of K-cycles

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 $\text{A}$ of a $K$–cycle stated in [\rfr(Conn4)], namely we show that a natural functional on $\text{A}$ is a trace functional. Then we discuss whether this functional gives a trace on the whole universal graded differential algebra $\text{Q}(\text{A})$. On the one hand we prove that the regularity conditions on $K$-cycles considered in [\rfr(Conn4)] imply the trace property on $\text{Q}(\text{A})$. 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.