Support of a faithful representation

Question

Let $A$ be a C*-algebra. Any representation $\pi:A\to B(H)$ is uniquely extended to a $w^*$-continuous representation $\tilde{\pi}:A^{**}\to B(H)$.

Q: I am looking for an example of a faithful representation $\pi$ such that $\tilde{\pi}$ is no longer faithful.

Answer 1

In most reasonable cases $\tilde{\pi}$ is not faithful.

Consider the diagonal embedding $\pi\colon \ell_\infty \to B(\ell_2)$ with respect to any orthonormal basis. Certainly there is no faithful representation $\ell_\infty^{**}\to B(\ell_2)$ as $\ell_\infty^{**}$ is not $\sigma$-finite. Consequently, $\tilde{\pi}$ is not faithful.

This can be generalised to the setting where $A$ acts on a separable Hilbert space and contains an element with uncountable spectrum.