[math] \newcommand{\mathds}{\mathbb}[/math]

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Prove that the finite dimensional [math]C^*[/math]-algebras are exactly the direct sums of matrix algebras

[[math]] A=M_{N_1}(\mathbb C)\oplus\ldots\oplus M_{N_k}(\mathbb C) [[/math]]

by decomposing first the unit into a sum of central minimal projections.