I'm trying to solve the following exercise from Paulsen's book "Completely bounded maps and operator algebras":
Here, $M_n$ is the $C^*$-algebra of complex $n \times n$ matrices.
I don't have a good way to attack this problem. But here is an observation.
- Since this is a simple algebra, a unital $*$-homomorphism is always injective, so we know that $M_n$ embeds in $B(\mathcal{K})$ via the representation $\pi$.
How can I construct the subspaces $\mathcal{H}_i$?
Best Answer
Complemented subspaces of subspaces of $\mathcal{K}$ correspond to orthogonal projection in $\mathcal{B}(\mathcal{K})$ (namely the orthogonal projection onto the subspace). In $M_n$, the matrices $E_{i,i}$ are projections.
Since $*$-homomorphisms take projections to projections, it follows that $\pi(E_{i,i})$ is a projection for each $i$. Then $\mathcal{H}_i := \pi(E_{i,i}) \mathcal{K}$ is a subspace of $\mathcal{K}$ and since $E_{i,i}E_{j,j} = \delta_{i,j}$ the subspace $\mathcal{H}$ are all mutually orthogonal. Moreover, since $\pi$ is unital $$ 1_{\mathcal{B}(\mathcal{K})} = \pi(1_{M_n}) = \pi(\sum_{i=1}^n E_{i,i}) = \sum_{i=1}^n \pi(E_{i,i}). $$ It follows that $\mathcal{K} = \bigoplus_{i=1}^n \mathcal{H}_i$. Since $E_{i,j}E_{k,l} = \delta_{j,k} E_{i,l}$ it follows that that for $\pi(E_{i,i})h \in \mathcal{H}_i$ we have $$ \pi(E_{j,i}) \pi(E_{i,i})h = \pi(E_{j,j}) \pi(E_{j,i}) h \in \mathcal{H}_j $$ so we may consider the restricted operator $\pi(E_{j,i}) \colon \mathcal{H}_i \to \mathcal{H}_j$, (really $\pi(E_{j,i})|_{\mathcal{H}_i}$) which has inverse $\pi(E_{i,j})$.
I'll leave the unitary equivalence statement to you, but think "change of basis". There's also some fiddling needs done to get this in the form of your original question.