At the moment I am trying to understand the proof of the multiplication operator version of the spectral theorem in "Operator Theoretic Aspects of Ergodic Theory" by Eisner, Farkas, Haase and Nagel (https://www.math.uni-leipzig.de/~eisner/book-EFHN.pdf) in Chapter 18.
In the proof they use the Gelfand-Naimark Theorem they explain earlier in the book. I have a few questions about the proof, since it is really short and has (for a student with not super much knowledge in functional analysis) not enough explanations to understand all the different steps...
My first question is about the isomorphisms in the proof below. I assume the first one comes from Gelfand-Naimark but I do not really understand how it works with this cyclic subspaces as it is explained above the proof. I also don't know why $\bigoplus_\alpha L^2(K, \mu_{x_\alpha}) \cong L^2(\bigcup_\alpha K_\alpha, \bigoplus_\alpha \mu_{x_\alpha})$ holds.
And as a bigger question: I also do not see where the assumptions such as the operator being bounded and normal are needed in the proof.
Thank you very much for helping!