TITLE: Cap product between K-Theory and K-Homology
QUESTION [5 upvotes]: In Exercise 9.8.9 of the book "Analytic K-Homology" by Higson and Roe one has to construct a cap product $K_p(A) \otimes K^q(A) \to K^{q-p}(A)$, if A is commutative.

Is the commutativity assumption on A really needed for the construction of this cap product?

I mean ... isn't the cap product just induced by the map $A \otimes \mathfrak{D}(A) \to \mathfrak{D}(A) / \mathfrak{K}$, $a \otimes T \mapsto [aT]$? (Or a slight variation of it accounting for the fact that we have to take the unitization of A somewhere.) This map is a $^\ast$-homomorphism regardless of the commutativity of $A$.

REPLY [2 votes]: Reposting my comment: in order for the map you describe to be defined it seems like you at least need $D(A)$ to contain $A$ (I don't see the point of the fraktur here; most of the time it just makes things harder to read). I don't see why this needs to hold if $A$ is noncommutative.