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retagged, removed unessential info from title, while changing "et al"
YCor
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Seeking to understand meaning of "von Neumann spectrum" in a paper of Bader–Furman–Shaker

In attempting to understand the paper "Superrigidity, Weyl groups, and actions on the circle" of Uri Bader, Alex Furman and Ali Shaker (linked at Furman's page)

I find that towards the end of the proof of Lemma 2.2 they have a situation where $G$ is a locally compact second countable group, $X$ is a Lebesgue space on which $G$ has a measure-class-preserving action, and $Y$ is a probability space on which $G$ has a measure-preserving action, and they speak of an isomorphim (a Banach space isomorphism, presumably) between $(L^{\infty}(Y))^{G}$ and $(L^{\infty}(X \times Y))^{G}$, and then say that this induces a Lebesgue space isomorphism between the corresponding von Neumann spectra, and I'm just wondering if anyone can clarify what "von Neumann spectrum" means in this instance. Presumably not just a subset of $\mathbb{C}$ associated with some Banach space operator. It seems as though it's a little difficult to resolve this question just by Googling.

Rupert
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