Some properties of operators (normal, self adjoint, hermitian) have intrinsic definitions for any element of a $C^*$-algebra. Is there such definition for compact operators?
Equivalently:
Let $\mathcal{A}$ be a $C^*$-algebra and $a \in \mathcal{A}$. Suppose that there's a representation of $\mathcal{A}$ in which $a$ is compact. Does it follow that in every irreducible representation of $\mathcal{A}$, $a$ will be compact?
No. It is possible that an element is mapped to a compact in one irreducible representation and to a non-compact in another.
For instance, let $I_n\in M_n(\mathbb C)$, $I\in B(\ell^2(\mathbb N))$ be the respectives identities. Let $\mathcal A=M_n(\mathbb C)\oplus B(\ell^2(\mathbb N))$ and $a=I_n\oplus I$. Then the irreducible representation $$ \pi_1:(x,y)\longmapsto x $$ maps $a$ to a compact, while the irreducible representation $$ \pi_2:(x,y)\longmapsto y $$ maps $a$ to a non-compact.