**EDIT**: André Henriques has commented below that the correct separability condition is not weak-\* separability as I have written below, but separability of the predual. 


This post came out a bit long, but if you're familiar with the topic, you can probably just skim through most of it - I've put the questions in bold. Some terminology:


1. A (commutative, as it shall always be in this post) $W^\* $-algebra is a $C^\* $-algebra which has a pre-dual as a Banach space. It is well known (see for example Takesaki's Theory of Operator Algebras I) that such algebras can be equivalently characterized as Von-Neumann subalgebras of the algebra of operators on a Hilbert space, or as spaces of the form $L^\infty \left (X, \mu \right) $ where $X$ is a locally compact space and $\mu$ a Radon measure.
2. By a standard Borel space I mean a measure (or sometimes just measurable) space which is Borel isomorphic to a complete separable metric space.


I am studying group actions on $W^\* $-algebras, and I am interested in particular in the question:



> 
>  **When can $ \left (X, \mu \right) $ above be chosen to be a standard Borel space?**  
> 
> 
> 


The question is important to me because if the answer is what I expect it to be, I have a very simple way of constructing concrete actions on compact measure spaces from given actions on the associated $W^\* $-algebra. (I can say more about the motivation, but I don't want to burden this post with too many details.)


My hypothesis is that the answer to my question is:  **Exactly when $A$ is separable in the weak-\* topology** . Certainly the $L^\infty $ of a standard Borel space is weak-\* separable: the Borel $\sigma$-algebra on a standard Borel space is countably generated, so rational linear combinations of the associated indicator functions are dense in $L^1$, making it norm-separable and hence its dual weak-\* separable. 


Here is how I tried to prove this: assume that $A$ is a weak-\* separable $W^\* $-algebra, and let $B$ be a norm-separable, norm-closed, weak-\* dense sub-$C^\* $-algebra (just take the norm-closed subspace generated by some weak-\* dense countable subset). I denote $A$ and $B$'s Gelfand spectra (i.e., the spaces of multiplicative linear functionals on these algebras) by $X\_A$ and $X\_B$. Since $A \simeq C(X\_A)$ and $B \simeq C(X\_B)$, I know that $X\_B$ is a complete separable metric space. $X$ above can be constructed as an open dense subset of $X\_A$ on which $\mu$ is supported. I want to prove that there is a null set $X\_A ^0$ such that $X\_A-X\_A ^0 $ is Borel isomorphic to $X\_B$. If I prove this, I win, because removing a null subset gives an isomorphic $L^\infty$. 


So basically, what I want to prove is: 



> 
>  **Given a $W^\* $-algebra $A$ and a weak-\* dense subalgebra $B$, there exists a null subset $X\_A ^0 $ such that $X\_A-X\_A ^0$ is Borel isomorphic to $X\_B$** . 
> 
> 
> 


It may be that I also have to remove a null subset of $X\_B$ - that's just as good for me, although I think it can be avoided. 


To prove the last statement, I tried going through the following: I have a natural map from $X\_A$ to $X\_B$ given by restricting a multiplicative functional to B. This map is definitely continuous (the topology on the Gelfand spectrum is given by pointwise convergence) and, while not a completely trivial fact, it is well known that it is onto (a multiplicative linear functional on a $C^\* $-subalgebra can always be extended to the entire algebra; see for example Kaniuth's  *A Course in Commutative Banach Algebras* , theorem 4.2.17). There's no reason, of course, to think that the restriction is one to one: extensions of functionals are in general highly non-unique. However, I believe that by using weak-\* density and taking away a null subset of $X\_A$, it can be made onto. That would already make the map a Borel isomorphism. The following observation may or may not be helpful: asking if there is a conull subset of $X\_A$ on which the restriction is one to one is the same as asking if there is a conull subset on which elements of $B$, viewed as functions, separate points.


Well, I could say more, but this post is already exceedingly long. I have been thinking and looking for information on these questions for quite some time now, so I would warmly welcome any suggestions or comments. 


To recap, my questions are:


1. When can $(X, \mu)$ be chosen to be a standard Borel space?
2. Is it true that 1 is equivalent to $A$ being weak-\* separable?
3. Is it true that the spectrum of a $W^\* $-algebra is always Borel isomorphic to the spectrum of a weak-\* dense subalgebra?



When does a $W^*$-algebra have a standard Borel spectrum?
[The category of commutative von Neumann algebras is contravariantly equivalent to the category of measurable spaces.](https://mathoverflow.net/questions/23408/reference-for-the-gelfand-neumark-theorem-for-commutative-von-neumann-algebras) Assuming the axiom of choice, isomorphism classes of objects in the above two categories that satisfy a certain countability property are classified by pairs (m,n)
of cardinal numbers, with the additional restriction
that either n=0 or n is infinite.
(Spaces that do not satisfy the countability property
cannot give separable von Neumann algebras.)


To a pair (m,n) the above correspondence assigns the measurable
space given by the coproduct of m isolated points (atoms)
and n copies of **R**.
Equivalently, one can take the product of m copies of the algebra **C** of complex numbers
and n copies of the von Neumann algebra of bounded measurable functions on the measurable space **R**.


Separability of the predual is equivalent to m and n being at most countable,
which also describes standard Borel spaces.


References for the classification of isomorphism classes of measurable spaces:


1) Theorem 3.4 in Irving Segal.
Equivalences of measure spaces.
American Journal of Mathematics 73:2 (1951), 275–313.
<http://dx.doi.org/10.2307/2372178>.


2) Theorem 1 in Dorothy Maharam.
On homogeneous measure algebras.
Proceedings of the National Academy of Sciences 28:3 (1942), 108–111.
<http://dx.doi.org/10.1073/pnas.28.3.108>


