# Finite-dimensional irreducible representations

How do we show that a finite-dimensional $*$-representation of a $C^{*}$-algebra is unitary equivalent to a direct sum of irreducible representations?

• Take orthogonal complements. – Qiaochu Yuan May 21 '13 at 18:29

If $M$ is a finite dimensional representation of a $C^*$ algebra $A$, then it is in fact a representation of a quotient $B=A/I$ by a closed ideal of finite codimension.
So you may just as well assume that $A$ is finite-dimensional. Then it is a direct product of matrix algebras, and it is therefore semisimple.