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

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

What you want follows at once from this.

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