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](http://web.me.com/ndh2/math/Papers_files/Higson,%2520Guentner%2520-%25202004%2520-%2520Group%2520C%2a-algebras%2520and%2520K-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 $D\_2$.



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


