I'm working on K-theory. Let H be an infinite dimensional separable Hilbert space and $A$ a $C^{\star}$-algebra. Let put $\mathcal{Q}(H):=\mathcal{B}(H)/\mathcal{K}(H)$ the Calkin algebra.
I've proved that the map $\delta$ : $K_{1}$($\mathcal{Q}(H)$)$\to$ $K_{1}$($\mathcal{Q}(H)$)$\simeq$ $\mathbb{Z}$ is given by the the index of a Fredholm operator.
Now i want to do the same thing with the Hilbert A-module $A\otimes H$. We have an exact sequence :
$0\longrightarrow \mathcal{K}(A\otimes H)\overset{i}{\longrightarrow} \mathcal{B}(A\otimes H){\overset{\pi}{\longrightarrow}} \:\mathcal{Q}(A\otimes H)\longrightarrow 0$
The six terms exact sequence gives us the isomorphism $\delta$ : $K_{1}$($\mathcal{Q}(A\otimes H)$)$\to$ $K_{0}(A)$.
My question is : how to explicit $\delta$ ?
I've seen in the Wegge-Olsen book, that we can generalized Fredholm operator in a Hilbert module but is there a easier way to define $\delta$ with the particular hilbert A-module $A\otimes H$ ?
Thanks !