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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.

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    $\begingroup$ I think I could answer this, but probably @UriBader.is the best person to answer. $\endgroup$ – Nik Weaver May 1 '20 at 14:51
  • $\begingroup$ @NikWeaver I guess you should answer the question then (I'd be uncomfortable to answer a question explicitly about my own work) $\endgroup$ – YCor May 1 '20 at 16:10
  • $\begingroup$ @YCor really? If someone had a question about one of your papers you would be uncomfortable answering? $\endgroup$ – Nik Weaver May 1 '20 at 18:25
  • $\begingroup$ @NikWeaver I mean answering here on MO; I'm happy with any kind of question on my papers or remarks (including errata as Uri recently did). Maybe "uncomfortable" is exaggerated, but MO is not a place for private communications, so it's good sign that another person is prone to answer. $\endgroup$ – YCor May 1 '20 at 18:37
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    $\begingroup$ Thanks, @NikWeaver and YCor, for calling my attention and for your care. I intend to leave an answer later on and clarify some things related to the question. Unfortunately I am unable to do it right now. In the meantime, anyone (Nik?) who wishes to answer the actual question asked, namely what does the term "von-Neumann spectrum" stands for, you are most welcome to do so. In fact, I do not have any reference handy currently. The definition is probably in Takasaki I, but most likely without giving this specific terminology. $\endgroup$ – Uri Bader May 2 '20 at 14:50
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Let me explain what we mean by the term "von Neumann spectrum".

Before doing so, let me recall the better known Gelfand duality: the functor $X\mapsto C(X)$, from the category of compact Hausdorff topological spaces to the category of unital commutative C*-algebras, establishes an equivalence of categories. The functor in the other direction is called the Gelfand spectrum. In particular, the compact space associated with a unital commutative C*-algebra $A$ under this functor is called the Gelfand spectrum of $A$.

Consider now the category of Lebesgue spaces. Here a Lebesgue space is a standard Borel space endowed with a measure class and a morphism of such is a class of a.e defined measure class preserving Borel maps, where two such are considered equivalent if they agree a.e. Recall also that a von Neumann algebra is a unital C*-algebra which is a dual space as Banach space. Let us say that a von Neumann algebra is separable if its predual (which is uniquely defined) is separable wrt the Banach space topology (note that an infinite dimensional von Neumann algebra is never separable as a Banach space, so this terminology should cause no confusion). There is an obvious functor from Lebesgue spaces to separable commutative von Nuemann algebras, $X\mapsto L^\infty(X)$. By the term von Neumann duality (which is not standard, but it should be) I refer to the fact that this functor establishes an equivalence of categories. The functor in the other direction is called the von Neumann spectrum. In particular, the Lebesgue space associated with a separable commutative von Nuemann algebra $A$ under this functor is called the von Neumann spectrum of $A$.

Again, let me stress that the term "von Neumann spectrum" is not completely standard, but it should be. I currently do not have a handy reference for the above discussion, but any book dealing with the foundations of the theory of von Neumann algebras should cover it. In particular, the above should be discussed when one decomposes a general von Neumann algebra as a direct integral of factors over its center. Note that the center is a commutative von Neumann algebra. The measured space carrying the direct integral is its spectrum.

Before concluding this explanation, I want to briefly expand on something that was not explicitly asked, but is very much related. It is a crucial fact that the variety of objects in the category of Lebesgue spaces is quite dull: up to isomorphisms there is a unique atomless Lebesgue space. But this doesn't mean the category itself is dull, only that its reachness in its morphisms. By analogy, think of the category of separable infinite dimensional Hilbert spaces, where again you have a unique class of objects but a reach variety of morphisms. So really, the miracle is in the group of automorphisms $\text{Aut}(X)$ and things get interesting when you consider representations of groups into this target. Studying these is (one aspect of) Ergodic Theory. It turns out that when one studies such representations of locally compact second countable groups, the von Neumann duality extends equivariantly. This fact is known by the slogan "Mackey's point realization theorem". But I guess I went far enough and end this discussion now.


Finally, let me make some comments about the paper under consideration and the relevant framework. First thing, you should know that this paper was not published, for some unfortunate circumstances. In particular, it was never refereed and it might have some rough edges. You should handle it with some care. Overall it is solid, but there might be some glitches in the presentation.

Second, the specific framework under which we considered lemma 2.2 was changed in time. We found that it is convenient to replace the "ergodic with unitary coefficient" assumption with the "metrically ergodic" assumption, defined in section 2 here. The latter is formally stronger, but we find it easier to handle (and generalize) in our later work. In particular, an analogue of lemma 2.2 is given in lemma 3.7 here.

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Uri may give a more detailed answer later, but a short answer is that "von Neumann spectrum" is also called "dynamical spectrum". (And the latter is easier to Google.)

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