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