About faithful representation of a finite-dimensional C*-algebra

Question

I know every C$^*$-algebra has a faithful representation on a Hilbert space. A representation of A is a $*$-homomorphism $\Phi: A \rightarrow B(H) $ for some Hilbert space $H$. Is there any special case when $A$ is a finite dimensional C$^*$-algebra, we cannot find a faithful representation of $A$ on an infinite dimensional Hilbert space?

Answer 1

If $A$ is finite-dimensional, doing GNS with a faithful state (the normalized trace, for instance) will give a faithful representation $\Phi:A\to B(H_0)$, with $H_0$ finite-dimensional. Now let $H=\bigoplus_{n\in\mathbb N}H_0,$ and define $\Phi_\infty:A\to B(H)$ by $$ \Phi_\infty(a)=\bigoplus_{n\in\mathbb N}\Phi(a). $$ Then $\Phi_\infty$ is a faithful representation of $A$ on an infinite-dimensional Hilbert space.