Suppose we have a $C^\*$ algebra $A=\{(x\_n)\in \prod M\_n(\Bbb C),lim\_n tr\_n(x\_n^\*x\_n)=0\}$.


If $B$ is any nonzero $C^\*$-sub algebra of $A$,does there exist a finite dimensional irreducible representation of $B$?



irreducible representation of a $C^*$ algebra
Yes. Irreducible representations are the building blocks. If there is a finite dimensional representation, there is a finite dimensional irrep. Any C\*-subalgebra of $A$, indeed any C\*-subalgebra of $\prod M\_n$, has plenty of finite dimensional representations, namely evaluation on the $n$th factor, for any $n$.


