# Products on the K-theory of graded C*-algebras

One can define products on the K-theory of graded C*-algebras as in http://web.me.com/ndh2/math/Papers_files/Higson,%20Guentner%20-%202004%20-%20Group%20C*-algebras%20and%20K-theory.pdf on page 152, but how can we show that when $C*$-algebra is unital and trivially graded, then the products agree with the product on the usual K_0 defines like $K_0(A)\otimes K_0(B)\rightarrow K_0(A\otimes B)$ by $[p]\otimes [q]=[p\otimes q]$. I think this is equivalent to show that the Fredholm index of $D_1\otimes 1+1\otimes D_2$ is the tensor product of the indices of $D_1$ and $D_2$.

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The cup-cap product of Kasparov, in bivariant $KK$-theory, is defined for graded $C^*$-algebras; for ungraded algebras, and when the first argument is $\mathbb{C}$, it restricts to the product you describe. See: Kasparov, G. G. Equivariant $KK$-theory and the Novikov conjecture. Invent. Math. 91 (1988), no. 1, 147–201.