Let $H$ be a separable Hilbert space and consider the Calkin algebra $C(H)=\frac{B(H)}{K(H)}$. 


Q) True or false: Any representation of $C(H)$ is a direct sum of irreducible representations. 



Representations of Calkin algebra
If you google "representations of the Calkin algebra", you find the 1967 paper of Sakai which answers your question in the negative: It says that the Calkin algebra has a type III factor representation. By the Lemma of Schur, however, every irrducible representation is a type I representation and a direct sum of more than one irreducible is not a factor.


