MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

One can define products on the K-theory of graded C*-algebras as in,%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 $D_2$.

share|cite|improve this question
up vote 2 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.