In attempting to understand this paper

 http://homepages.math.uic.edu/~furman/preprints/supercircle061406.pdf

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.

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.