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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 $A$.

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