This is an exercise (4.11) from the book Completely Bounded Maps and Operator Algebras by Vern Paulsen.
Let $\pi:M_n\to\mathcal{B}(K)$ be a unital $*$-homomorphism. Prove that, upto equivalence, $K=H_1\oplus\cdots\oplus H_n$, with $H_i=H\ \forall i=1,\ldots,n$, such that $\pi(E_{i,j})$ is the identity map from $H_j$ to $H_i$. Show that, upto equivalence, $\pi$ is the direct sum of $\text{dim}(H)$ copies of the identity map.
Using this exercise, he wrote as a remark (pg no. 50) that if $\pi:M_n\to\mathcal{B}(K)$ is a $*$- homomorphism, then up to unitary equivalence, $K$ decomposes as an orthogonal direct sum on $n$-dimensional subspaces such that $\pi$ is identity representation on each of the subspace.
I'm much confused with the exercise and the above remark. As per the remark if $K=H_1\oplus\cdots\oplus\cdots\oplus H_k$ such that $\dim(H_i)=n$ and $\pi|_{H_i}$ is identity representation, then shouldn't that $\pi$ be identity representation of $M_n$?
And in the exercise, it is asked to show that $\pi$ is direct sum of $\text{dim}(H)$ copies. By the first part $K$ is direct sum of $"n"$ copies. Then is $n=\text{dim}(H)$ and $\text{dim}(K)=n^2$/ Moreover, why $K$ is finite dimensional?
Can anyone help me with any idea to understand and solve the exercise and how the remark follows from that exercise? Thanks for your help in advance.
Edit: I have understood the meaning of the remark. It says up to unitary equivalence, $K=\sum\limits_{i\in I}\oplus \mathbb{C}^n_i$ and if $P_i$ denotes the projection of $K$ onto $\Bbb{C}^n_i$, then $P_i\pi(A)|_{\Bbb{C}^n_i}=A$ for each $i\in I$.