Writing a Hilbert C*-submodule of L2(X)L^2(X) as an integral sum over Hilbert subundles

In Ergodic Theory, some (though not all) presentations of compact extensions use Hilbert bundles.
Given (X,B,μ,T) and a sub σ-algebra GA one has an associated factor map π:(X,μ,T)(Y,ν,T) (where π(x)=π(x) iff (xAxA) for every AG) and a decomposition μ=μydν (where ν=π(μ)).

Then one can informally think of the space of functions in L2(X,μ) as an “integral” over the spaces of functions Hy:=L2(X,μy) for ν-a.e. y. Moreover, the union yHy defines a Hilbert bundle over Y (of course, Y a priori has no topological structure, so this is not strictly precise).
This is the approach taken in these notes by Yuri Lima and in Ch. 9 of Glasner’s book, for instance.

One then considers L(Y)-submodules (i.e. closed L(Y,ν)-invariant subspaces) of L2(X,μ). Now, here comes my problem. In proving many of the properties that one wants to prove for any such submodule M (I won’t go into these details or into compact extensions themselves) the authors usually consider the collection of {My}yY, which are thought of as closed subspaces of the Hy defined above (i.e. {My}yY is a subbundle of the Hilbert bundle). If fM, then fy is in My, where fy is just f “seen” as an element of L2(X,μy) instead of L2(X,μ) (warning: this is not the conditional expectation μy(f)). For instance, if we fix topological models such that X=Y×U then fy=f(y,) The problem is that f is not a function, it is an equivalence class of functions. Certainly, for a fixed fL2(Y×U) the function fy=f(y,) is defined for ν-a.e. y. But that doesn’t mean we can actually define a map M{My}ν-a.e.y for any closed subspace M, since such subspaces have uncountably many f‘s.

Question: Is there a way around this technical problem?

Answer

Attribution
Source : Link , Question Author : Moritz , Answer Author : Community

Leave a Comment