Suppose that $x$ is an element of abstract $C^*$-algebra $A$. For example if $x$ is normal, i.e. $x^*x=xx^*$ then if we use any representation $\pi$ of $A$ on some Hilbert space $H$ then $\pi(x)$ will be normal and therefore its index will be zero. I would like to know whether the following is true:
Suppose that $x$ is an element of a $C^*$-algebra $A$ and $\pi$ is a faithful representation on $H$ such that $\pi(x)$ is Fredholm. Is it true that the index of $\pi(x)$ is independent from the choice of such $\pi$?
No. For instance, you can take the representation $\pi\oplus\pi$ on $H\oplus H$, and then $(\pi\oplus\pi)(x)$ will have twice the index of $\pi(x)$.