Return to Revisions

The morphism $f : A \to A \otimes \mathbb K$ which maps $a$ to $a \otimes e_{11}$ induce isomorphism $K_1(f) : K_1(A) \to K_1(A \otimes \mathbb K)$ on $K_1$ groups as well. In fact, we can construct very explicit inverse $KK_0(A \otimes \mathbb K, A)$ cocycle. Indeed, suppose $\mathbb K$ acts on separable Hilbert space $\mathcal H$, then $A \otimes \mathcal H$ is naturally Fredholm $A \otimes \mathbb K, A$ bimodule, which is naturally cocycle inside $KK_0(A \otimes \mathbb K, A)$. It's not so hard to see that Kasparov product with the morphism above is an identity.