Definition: A $C^\*$-algebra $A$ is called sub-homogeneous if there exists $n\in\mathbb{N}$ such that every irreducible representation of $A$ has dimension at most $n$.


I could not find a proof or a counter example for the following claim:


Let $A,B$ be sub-homogeneous $C^\*$-algebras and let $C$ be a $C^\*$-algebra s.t. there is an exact sequence: $0\to A\to C\to B\to 0$.
Does it follow that $C$ is sub-homogeneous?


Thanks



Extensions of sub-homogeneous $C^*$-algebras are subhomogeneous
Yes. Since $C^{\*\*}\cong A^{\*\*}\oplus B^{\*\*}$, uniform bounds on the dimension of the irreducible representations of $A$ and $B$ impose a uniform bound on the dimension of the irreducible representations of $C$


