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Let $A$ be a $C^*$-algebra, $A$ finite dimensional. Then there is a faithful, non-degenerate representation of $A$.

How to prove it?. Take an irreducible representation $\pi_1:A\to L(H_1)$ of $A$ with $H_1=\overline{\pi(A)x}$ for a fixed $x\in H_1\setminus\{0\}$. It is $\dim\pi(A)<\infty$, but is $\dim H_1<\infty$?

The rest is clear now, I still have to construct a representation which is faithful.

banach-c
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  • Are you thinking of this as a step towards proving that every finite-dimensional C-algebra is (isomorphic to) one of the form $\bigoplus_{i=1}^N M_{n_i}(\mathbb{C})$? If you already know that finite-dimensional C-algebras look like this, then it is pretty clear how to get a faithful representation, nondegenerate representation.... – Mike F Sep 27 '15 at 17:27
  • Thanks for your comment. No, I still don't know that, the statement is new for me. I only try to construct a representation which is faithful. – banach-c Sep 27 '15 at 18:03

1 Answers1

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$H_1$ is obviously finite-dimensional: if $a_1,\ldots,a_n$ is a basis for $A$, then $$ \pi(a_1)x,\ldots,\pi(a_n)x $$ span $H_1$.

Martin Argerami
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