Decomposition of VN(G) into direct sum of matrix algebras when G is a compact group. Does anyone know any concrete proof of the following statement which is from the accepted answer to this question? Or any resource that has the proof?
When $G$ is a compact group, by using classical Peter-Weyl theory, it follows that $VN(G)$ is a direct sum of matrix algebras, corresponding to irreducible unitary representations of G.
 A: I'll take these notes as a reference, Theorem 17 in particular. (2) states that the left-regular representation $\lambda$ of $G$ is unitarily equivalent to
$$
\bigoplus_{(\pi,H_\pi)}\pi^{\oplus d_\pi},
$$
where the direct sum runs over a set of representatives of the equivalence classes of irreducible representations of $G$ and $d_\pi$ is the dimension of $H_\pi$. The latter is finite by (1). Thus
$$
\mathrm{VN}(G)\cong \left(\bigoplus_{(\pi,H_\pi)}\pi^{\oplus d_\pi}(G)\right)^{\prime\prime}\subset B\left(\bigoplus_{(\pi,H_\pi)}H_\pi^{\oplus d_\pi}\right).
$$
It is elementary to see that
$$
\left(\bigoplus_{(\pi,H_\pi)}\pi^{\oplus d_\pi}(G)\right)^{\prime\prime}=\bigoplus_{(\pi,H_\pi)}(\pi(G)^{\prime\prime})^{d_\pi}.
$$
Note that the direct sums here are $\ell_\infty$-direct sums, i.e., contain all sequences of operators with uniformly bounded norm.
Moreover, as $\pi$ is irreducible, $\pi(G)^\prime=\mathbb C I$ (see Theorem 12, also known as Schur's lemma), and hence $\pi(G)^{\prime\prime}=B(H_{d_\pi})\cong M_{d_\pi}(\mathbb C)$. Therefore
$$
\mathrm{VN}(G)\cong\bigoplus_{(\pi,H_\pi)}M_{d_\pi}(\mathbb C)^{d_\pi}.
$$
