Let $A$ be a simple $C^*$-algebra. I am trying to prove that $A$ admits a non-zero finite dimensional representation if and only if $A\cong M_n(\mathbb{C})$ for some $n$.
The reverse implication is trivial. For the other one, if $\varphi:A\to B(\mathbb{C}^n)$ is a non-zero finite dimensional representation of $A$, then $\varphi$ is faithful, because $A$ is simple. Since $B(\mathbb{C}^n)\cong M_n(\mathbb{C})$, we have that $A$ is isomorphic to a simple $*$-subalgebra of $M_n(\mathbb{C})$. This is as far as I can go. Any ideas on how to go on?
P.S: I have seen a proof using vN algebras, but the thing is I came across this exercise in a book before the chapter on vN algebras, so I am trying to solve this without vN algebras (or irreducible representations).
Also: I know the classification theorem of finite dimensional $C^*$-algebras, but I can't use this. I want to prove this result in order to classify finite dimensional $C^*$-algebras.
Best Answer
You have to assume $\varphi$ is non-degenerate (i.e., unital). Otherwise you need to restrict the codomain.
Once you have that $\varphi$ is unital, all you need is to consider a unital, simple, C$^*$-subalgebra of $M_n(\mathbb C)$; I will still call it $A$. Consider the center $Z(A)$ of $A$. This is a finite-dimensional, abelian, C$^*$-algebra. Use the Spectral Theorem or Functional Calculus to show that $A$ has a projection; then it has a minimal projection $p$. Now, because $p\in Z(A)$, the subalgebra $Ap$ is an ideal; as $A$ is simple, $p=I$. Thus $Z(A)=\mathbb CI$.
Now use again the Spectral Theorem or Functional Calculus to get a projection $p\in A$; and again since $\dim A<\infty$, there exists a minimal projection $p_1\in A$. If $p_1A(I-p_1)=0$, then for any $a\in A$ we have $$\tag1 p_1a=p_1ap_1+p_1a(I-p_1)=p_1ap_1. $$ If $a=a^*$, taking adjoints in $(1)$ then $p_1a=ap_1$. As selfadjoint elements span the whole algebra, we get that $p_1\in Z(A)$; this would imply $p_1=I$, which is only possible when $n=1$. It follows that $p_1A(I-p_1)\ne0$: that is, there exists $a\in A$ such that $p_1a(I-p_1)\ne0$. Let $vr=p_1a(I-p_1)$ be the polar decomposition.
Note that, as $p_1$ is minimal, the range of $p_1$ agrees with the range of $p_1a(I-p_1)$. Then $v^*v=p_1$. Define $p_2=vv^*$. Note that $v=p_1v(I-p_1)$, so $p_1p_2=v^*vvv^*=0$. Name $v=v_{1}$. Repeat the procedure, now on the algebra $(I-p_1)A(I-p_1)$, and starting with $p_2$, to obtain a minimal projection $p_3\in (I-p_1)A(I-p_1)$ with $p_3p_2=0$ and with a partial isometry $v_{2}$ such that $v_{2}^*v_{2}=p_2$, $v_{2}v_{2}^*=p_3$. As $A$ is finite-dimensional, the process finishes and we end up with pairwise orthogonal minimal projections $p_1,\ldots,p_k$, and partial isometries $v_{s}$, $s=1,\ldots,k-1$, such that $v_{s}^*v_{s}=p_s$, $v_{s}v_s^*=p_{s+1}$. Define $$ E_{rr}=p_r,\ \ E_{1r}=v_{r-1}v_{r-2}\cdots v_1. $$ Then $$ E_{1r}^*E_{1r}=p_1,\ \ \ E_{1r}E_{1r}^*=p_r. $$ Next define $$ E_{r1}=E_{1r}^*,\ \ \ E_{rs}=E_{r1}E_{1s}. $$ It is then easy to check that $$\tag2 E_{rs}E_{vw}=\delta_{sv}\,E_{rw},\ \ \ E_{rs}^*=E_{sr}. $$ It is now straightforward to check that the map $\phi:M_k(\mathbb C)\to A$, given by $[a_{rs}]\longmapsto \sum_{rs}a_{rs}E_{rs}$ is a $*$-isomorphism. Thus $A\simeq M_k(\mathbb C)$.
Note that you cannot expect $k=n$ in general. For instance, you can embed $M_2(\mathbb C)$ as a unital $*$ subalgebra of $M_4(\mathbb C)$ by $$ \begin{bmatrix} a&b\\ c&d\end{bmatrix} \longmapsto \begin{bmatrix} a&0&b&0\\ 0&a&0&b\\ c&0&d&0\\0&c&0&d\end{bmatrix} . $$