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A measurable field of Hilbert spaces is the exact analogue of a vector bundle over a topological spaces in the setting of bundles of infinite-dimensional Hilbert spaces over measurable spaces.
The original definition is due to John von Neumann (Definition 1 in \cite{Neumann}).
We present here a slightly modernized version, which can be found in many modern sources, e.g., Takesaki \cite{Takesaki}.
\begin{definition} Suppose $(X,\Sigma)$ is a measurable space equipped with a σ-finite measure $\mu$, or, less specifically, with a σ-ideal $N$ of negligible subsets so that $(X,\Sigma,N)$ is an enhanced measurable space. A measurable field of Hilbert spaces over $(X,\Sigma,N)$ is a family $H_x$ of Hilbert spaces indexed by points $x\in X$ together with a vector subspace $M$ of the product $P$ of vector spaces $\prod_{x\in X} H_x$. The elements of $M$ are known as measurable sections. The pair $(\{H_x\}_{x\in X},M)$ must satisfy the following conditions. * For any $m\in M$ the function $X\to\mathbf{R}$ ($x\mapsto \|m(x)\|$) is a measurable function on $(X,\Sigma)$. * If for some $p\in P$, the function $X\to\mathbf{C}$ ($x\mapsto\langle p(x),m(x)\rangle$) is a measurable function on $(X,\Sigma)$ for any $m\in M$, then $p\in M$. * There is a countable subset $M'\subset M$ such that for any $x\in X$, the closure of the span of vectors $m(x)$ ($m\in M'$) coincides with $H_x$. \end{definition}
The last condition restrict us to bundles of separable Hilbert spaces. One can also define bundles of nonseparable Hilbert spaces, but this cannot be done simply by dropping the last condition.
The category of measurable fields of Hilbert spaces on $(X,\Sigma,N)$ is equivalent to the category of W*-modules over the commutative von Neumann algebra $\mathrm{L}^\infty(X,\Sigma,N)$.
(If we work with bundles of separable Hilbert spaces, then W*-modules must be countably generated.)
\bibitem{Neumann} John Von Neumann, On Rings of Operators. Reduction Theory, The Annals of Mathematics 50:2 (1949), 401. doi.
\bibitem{Takesaki} Masamichi Takesaki, Theory of Operator Algebras. I, Springer, 1979.
How do you reconcile the statements
The last condition restrict us to bundles of separable Hilbert spaces. One can also define bundles of nonseparable Hilbert spaces, but this cannot be done simply by dropping the last condition.
and
(If we work with bundles of separable Hilbert spaces, then W*-modules must be countably generated.)
Isn’t the definition currently given in a way such that the fibres are always separable? I would think the better way to state the Serre-Swan type duality is by default with countably-generated W*-modules, then add a comment about the generalisation to the non-separable case.
Re #3: Yes, either both fibers and modules are countably generated, or not.
Reworked the Serre–Swan material to say
The category of measurable fields of Hilbert spaces on $(X,\Sigma,N)$ (as defined above) is equivalent to the category of countably-generated W*-modules over the commutative von Neumann algebra $\mathrm{L}^\infty(X,\Sigma,N)$.
(If we work with bundles of general, possibly nonseparable Hilbert spaces, then the W*-modules do not need to be countably generated.)
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