A remark on trace properties of K-cycles
Abstract
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.