I'm working through the proof of Plancherel's Theorem in $C^{*}$-algebras by Dixmier, section 18.8. For the most part, I'm happy with it, although I have one problem. From Dixmier, I have the proof that
$L^{2}(G)\cong\int_{\widehat{G}}^{\oplus}{(K(\pi)\otimes\overline{K}(\pi))\ d\mu(\pi)},$
where $\pi\mapsto K(\pi)$ denotes the canonical field of Hilbert spaces. That is, if for $n\in\Bbb{N}\cup\{\infty\}$, we denote by $\widehat{G}_{n}$ the set of elements of dimension $N$ in $\widehat{G}$, then $\pi\mapsto K(\pi)$ reduces on $\widehat{G}_{n}$ to the standard $n$-dimensional Hilbert space.
I got into reading this proof from Principles of Harmonic Analysis by Anton Deitmar, which gives the statement of the equivalent part of the Plancherel Theorem as
$L^{2}(G)\cong\int_{\widehat{G}}^{\oplus}{\mathrm{HS}(V_{\pi})\ d\mu(\pi)},$
and says that the proof is in Dixmier.
Then my question is, how does one see that these two are indeed equivalent? Clearly, it suffices to show that
$\mathrm{HS}(V_{\pi})\cong K(\pi)\otimes\overline{K}(\pi),$
but I simply have no idea how to go about showing that. I suspect that I'm just being stupid and missing something which makes it particularly obvious, which probably stems from a lack of understanding of the canonical field of Hilbert spaces over $\widehat{G}$. I'm reasonably happy with the definition, but I don't really know how to make use of it here. The only real attempt I've been able to come up with is based on the idea of writing $\widehat{G}=\bigoplus\widehat{G}_{n}$, and then
$\int_{\widehat{G}}^{\oplus}{(K(\pi)\otimes\overline{K}(\pi))\ d\mu(\pi)}=\bigoplus_{i}\int_{\widehat{G}_{n}}^{\oplus}(K(\pi)\otimes\overline{K}(\pi))\ d\mu(\pi)=\bigoplus_{i}\int_{\widehat{G}_{n}}^{\oplus}{(H_{i}\otimes\overline{H_{i}})\ d\mu(\pi)},$
where $H_{i}$ is the standard $n$-dimensional Hilbert space. I'm aware that for finite $n$, this is just $\Bbb{C}^{n}$, although I am at a complete loss as to what Dixmier means by it for $H_{\infty}$ (if he means anything by it, he might not intend for you to look at it this way...). Even ignoring that issue, I'm at a loss as to where I should go after that (if what I wrote is even true, I'm not convinced you can split up the direct integral into a direct sum of integrals the way I did).
Any help either with the proof of the equivalence of the two Theorems, or my interpretation of $\pi\mapsto K(\pi)$ would be much appreciated!