how to prove $\prod_{i\in{I_j}}\mathbb{M}_j(\mathbb{C})\cong\mathbb{M}_j(l^\infty(I_j))$

Question

I try to read the book of C*-algebra and finite- dimensional approximations. In the proof of Theorem 2.7.7, I don't know how to prove $\prod_{i\in{I_j}}\mathbb{M}_j(\mathbb{C})\cong\mathbb{M}_j(l^{\infty}{I_j}))$, where {${I_j:j\in N}$ (natural number)} is the set of pure states with J-dimensional GNS representaitons and $\pi_j:\mathcal{A}\to\prod_{i\in{I_j}}\mathbb{M}_j(\mathbb{C})$. Thank you!!!

Answer 1

It is just the general assertion that $$ \prod_{k\in K}M_n(\mathbb C)\simeq M_n(\ell^\infty(K)). $$ It is almost just notation. Given a sequence $\{X_k\}\in \prod_kM_n(\mathbb C)$, define $\varphi(\{X_k\})$ to be the $n\times n$ matrix such that its $i,j$ entry is the sequence $\{(X_{k})_{ij}\}\in\ell^\infty(K)$. That the sequence is bounded follows from the fact that $\|\{X_k\}\|=\sup_k\|X_k\|$ is finite. That $\varphi$ is a $*$-isomorphism is now routine.