In a C*-algebras paper I've read recently, they state that every irreducible representation of the C*-algebra $C(X,\mathbb{C})$, where X is compact topological space, is 1-dimensional.
They also state that every irreducible representation of $C(X,M_{n}(\mathbb{C}))$ is n-dimensional. (where again X is compact and $M_{n}(\mathbb{C})$ is C*-algebra of complex n*n matrices)
They state these without proofs. Could you please help me how to prove these things, or tell me literature in which I can find proof/insight?
Thanks.
In the case of $C(X)$, this is an abelian algebra, so the image under a representation $\pi:C(X)\to B(H)$ will be abelian. But $\pi(C(X))$ is wot dense in $B(H)$ (by the irreducibility) so $B(H)$ is abelian: thus $\dim H=1$.
Note that $C(X,M_n(\mathbb C))=M_n(C(X))$ (properly, there is an identification to be made, but if you think about it, in both cases you have matrices of functions). Now let $\pi:M_n(C(X))\to B(H)$ be irreducible. As $X$ is compact, $C(X)$ is unital. So the matrix units $\{E_{kj}\}$ map to a system of matrix units $\{F_{kj}\}$ in $B(H)$. We can form a map $$\pi_0: C(X)\to M_n(C(X))\to F_{11} B(H)F_{11} $$ by $\pi_0(f)= \pi(f\otimes E_{11})$ (where $f\otimes E_{11}$ is the matrix with $f$ in the $1,1$ corner and zeroes everywhere else). It is straighforward to check that $\pi_0$ is a representation. It also happens that $\pi_0$ is irreducible (where we identify $F_{11}B(H)F_{11}$ with $B(F_{11}H)$): for any $T=F_{11}TF_{11}\in F_{11}B(H)F_{11}$, by hypothesis there exists a net $\{A_j\}\subset M_n(C(X))$ with $\pi(A_j)\to T$ wot. We also have $$ T=F_{11}TF_{11}=\lim_jF_{11}\pi(A_j)F_{11}=\lim_j\pi(E_{11}A_jE_{11}). $$ So, with the identification $C(X)\simeq E_{11}M_n(C(X))E_{11}$, we have found a net $\{f_j\}\subset C(X)$ such that $\pi_0(f_j)\to T$; thus $\pi_0$ is irreducible. Now using that $C(X)$ is abelian, we conclude that $F_{11}H$ is one dimensional. So $F_{11}$ is a rank-one projection; and $F_{kk}=F_{1k}^*F_{1k}$ is equivalent to $F_{11}=F_{1k}F_{1k}^*$, so it is also one-dimensional. Then $H=\mathbb C^n$ and $B(H)=M_n(\mathbb C)$.